From ffe1757686d2897b2e5f76ac38864915e6c2ef8e Mon Sep 17 00:00:00 2001 From: jvivian Date: Mon, 14 Oct 2024 22:37:29 -0700 Subject: [PATCH 01/15] Initial post, classical mechanics --- posts/classical-mechanics/index.qmd | 374 ++++++++++++++++++++++++++++ 1 file changed, 374 insertions(+) create mode 100644 posts/classical-mechanics/index.qmd diff --git a/posts/classical-mechanics/index.qmd b/posts/classical-mechanics/index.qmd new file mode 100644 index 0000000..e043406 --- /dev/null +++ b/posts/classical-mechanics/index.qmd @@ -0,0 +1,374 @@ +--- +title: "Pythagorean Time Dilation" +subtitle: "A Summary of Classical Mechanics" +author: John Vivian +date: 2024-10-15 +format: + html: + # css: styles.css + toc: true + toc-depth: 2 + echo: false +editor: visual +execute: + render-on-save: true +tools: + - mermaid +--- +```{python} +import numpy as np +import matplotlib.pyplot as plt +import seaborn as sns + +plt.style.use('ggplot') +``` + +# Acceleration is Absolute +> _I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me_ [^1] + +[^1]: Isaac Newton. "Anecdotes, Observations and Characters, of Books and Men" Vol 1 (page 158). Spence, J. 1820. + +Imagine yourself in space with no nearby objects, only distant stars; would it be possible to tell you are moving? Nope. As Einstein later proved, velocity only has context relative to another reference frame. Two people moving in in the same environment past each other would not be able to conclude whether one person was stationary and the other was moving or if both people are moving. + +Oddly, _acceleration_ is not the same. If you were suddenly accelerated while in the space suit you would feel it; you would jerk forward, feel the suit push against your chest, etc. The reason for this is acceleration produces a _force gradient_, as opposed to velocity which does not require a force to keep you moving (an object in motion, yada yada...). + +You can build your own accelerator-detector just by hanging a weight from a spring. If you begin accelerating left or right, the spring would stretch in the opposite direction. The weight needs to hang to work, because if you turn it sideways the weight will fall towards the ground, even if remain perfectly still. You _are_ accelerating, and have been for quite some time, directly towards the center of the planet. We experience this acceleration as _weight_, which is canceled out by the normal force of the planet pushing back up against us. + + +```{python} +#| label: fig-spring_weight_visualization +#| fig-cap: "Spring with hanging weight, which stretches to the left if accelerated to the right." +# Parameters for the spring and weight +k = 50 # Spring constant (N/m) +mass = 1.0 # Mass of the weight (kg) +g = 9.81 # Gravitational acceleration (m/s^2) +equilibrium_length = 2.0 # Length of the spring with no weight (m) + +# Calculate the displacement due to the weight +stretch = mass * g / k # Stretch caused by the weight + +# Create the spring curve using sine waves for aesthetics +def spring_curve(x: np.ndarray, stretch: float) -> np.ndarray: + """Creates a wavy curve representing the spring.""" + return np.sin(10 * np.pi * x) * 0.1 - x * stretch + +# Generate the spring curve before and after stretching +x = np.linspace(0, 1, 500) # x-axis points along the spring +y_unstretched = spring_curve(x, 0) # Unstretched spring +y_stretched = spring_curve(x, stretch) # Stretched spring + +# Create the figure and axis +fig, ax = plt.subplots(figsize=(6, 8)) + +# Plot the unstretched and stretched spring curves +ax.plot(y_unstretched, -x * equilibrium_length, 'k--', alpha=0.5, label='Unstretched Spring') +ax.plot(y_stretched, -x * (equilibrium_length + stretch), 'b-', label='Stretched Spring') + +# Plot the weight as a black circle +ax.plot(0, -(equilibrium_length + stretch), 'ro', markersize=15, label='Weight') + +# Plot the gravitational force vector (downward) +ax.arrow(0, -(equilibrium_length + stretch), 0, -0.5, head_width=0.1, head_length=0.1, + fc='blue', ec='blue', label='Gravitational Force $F_g$') + +# Plot the spring force vector (upward) +ax.arrow(0, -(equilibrium_length + stretch), 0, 0.5, head_width=0.1, head_length=0.1, + fc='green', ec='green', label='Spring Force $F_s$') + +# Set plot limits and labels +ax.set_xlim(-0.5, 0.5) +ax.set_ylim(-3.5, 0.5) +ax.set_xlabel('Spring Width') +ax.set_ylabel('Height') +# ax.set_title('Spring with Hanging Weight Demonstrating Acceleration') + +# Add legend +ax.legend(loc='upper right') + +# Display the plot +plt.show() +``` + +Physics equations are incredible because they describe _universal relationships_ about the world we've found ourselves in. For example, the _momentum_ of an object is described by its mass and velocity: $p = mv$. This relationship applies to _every_ object of mass that can have a velocity. And I think that's pretty neat [^2]. This equation also let's us say momentum is proportional to velocity given constant mass. If you double the velocity of an object, you double its momentum. + +[^2]: Neature Walk: https://www.youtube.com/watch?v=Hm3JodBR-vs + +## Kinematic Equations +> Moving Through Time: Universal Relationships of Motion + +These describe universal relationships about an object's motion with constant acceleration—useful when you want to predict where or how fast something will be at a given time. + +$$v = v_0 + at$$ + +Velocity increases linearly with time when acceleration is constant + +$$s = s_0 + v_0 t + \frac{1}{2} at^2$$ + +Position grows with time, influenced by both your initial speed and the acceleration + +$$v^2 = v_0^2 + 2a(s - s_0)$$ + +The relationship between velocity and distance, without needing time + +An interesting consequence to these equations is given position or velocity as a function of time, the equation will predict the function's value for whatever $t$ is. Meaning, these universal relationships do not operate as if the past (fixed from our perspective) is fundamentally different from the future (not-fixed, from our perspective). + +::: {.callout-tip collapse="true"} +## What's the point of calculus +Calculus solves two limit-based problems: + +1. The goal of the derivative is to find $\frac{dx}{dt}$ as they both approach zero, but you can't divide by zero. +2. The integral attempts to find the sum of the area under the curve as the number of riemann squares approaches infinity. + +You can expand a small change in a simple function (such as $x^2$, i.e. $(x+h)^2 - x^2$) and work out as the limit $h \rightarrow 0$, you get $2x$. +::: + +:::{.callout-note collapse="true"} +## Kinematic Derivation +The kinematic equations can be derived from the basic definition of calculus. Given a change in position over time is velocity: +$$\frac{dx}{dt} = v$$ + +we want to express $v$ in terms of $t$, since that also changes over time. + +$$v(t) = v_0 + at$$ + +Reformulate velocity as the first time derivative of position + +$$\frac{dx}{dt} = v_0 + at$$ + +Then integrate with respect to time + +$$x(t) = \int (v_0 + at) dt$$ + +Solving for the integral gives us + +$$x(t) = v_0t + \frac{1}{2}at^2 + C$$ + +Where C would represent whatever the **initial position** is, which we can call $x_0$. This gives us + +$$x(t) = x_0 + v_0t + \frac{1}{2}at^2$$ + +You can plug this equation into $v = v_0 + at$ and do some math to get: + +$$v^2 = v_0^2 + 2a(x-x0)$$ +::: + +# Lagrangian and Hamiltonian Mechanics +> Different ways to tell the same (Newtonian) story + +Instead of forces and acceleration, Lagrangian and Hamiltonian mechanics focuses on _paths, energy, and phase space_. The last is particularly useful for understanding field theory. + +```{python} +#| label: fig-corrected_mechanics_comparison +#| fig-cap: "Comparison of Newtonian, Lagrangian, and Hamiltonian Mechanics" +from scipy.integrate import solve_ivp +# Parameters for the harmonic oscillator +m = 1.0 # Mass (kg) +k = 1.0 # Spring constant (N/m) +omega = np.sqrt(k / m) # Angular frequency +A = 1.0 # Amplitude of oscillation + +# Newtonian mechanics: F = ma = -kx +def newtonian_acceleration(x: np.ndarray) -> np.ndarray: + return -omega**2 * x + +# Hamiltonian mechanics: Phase space dynamics +def hamiltonian_system(t: float, y: np.ndarray) -> np.ndarray: + q, p = y # Position and momentum + dqdt = p / m # dx/dt = p / m + dpdt = -k * q # dp/dt = -kx + return [dqdt, dpdt] + +# Solve Hamiltonian system for phase space trajectory +sol = solve_ivp(hamiltonian_system, [0, 10], [A, 0], t_eval=np.linspace(0, 10, 500)) + +# Lagrangian path: Position over time assuming maximum displacement at t=0 +time = np.linspace(0, 10, 100) +x_lagrangian = A * np.cos(omega * time) # Expected path for least action + +# Create figure with subplots +fig, axs = plt.subplots(3, figsize=(8, 10)) + +# 1. Newtonian mechanics: Acceleration vs. Position +x = np.linspace(-A, A, 100) # Position range for the oscillator +axs[0].plot(x, newtonian_acceleration(x), color='blue') +axs[0].set_title('Newtonian Mechanics: F = ma') +axs[0].set_xlabel('Position ($x$)') +axs[0].set_ylabel('Acceleration ($a$)') +axs[0].axhline(0, color='black', lw=0.8) # Equilibrium acceleration line + +# 2. Lagrangian mechanics: Path over time +axs[1].plot(time, x_lagrangian, color='green') +axs[1].set_title('Lagrangian Mechanics: Path Minimization') +axs[1].set_xlabel('Time ($t$)') +axs[1].set_ylabel('Position ($x$)') +axs[1].axhline(0, color='black', lw=0.8) # Equilibrium line + +# 3. Hamiltonian mechanics: Phase space trajectory +axs[2].plot(sol.y[0], sol.y[1], color='red') +axs[2].set_title('Hamiltonian Mechanics: Phase Space') +axs[2].set_xlabel('Position ($q$)') +axs[2].set_ylabel('Momentum ($p$)') +axs[2].axhline(0, color='black', lw=0.8) # Zero momentum line +axs[2].axvline(0, color='black', lw=0.8) # Zero position line + +# Adjust layout and display plot +plt.suptitle('Comparison of Newtonian, Lagrangian, and Hamiltonian Mechanics') +plt.tight_layout() +plt.show() + +``` + +## The Action Integral +> The profound economy of Nature + +The action integral $S$ is the "score" of a given path taken by a system between two points in time. + +$$S[q(t)] = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt$$ + +- $S[q(t)]$ is the _action_ which depends on the path $q(t)$ +- $L(q, \dot{q}, t)$ is the **Lagrangian**, which is the difference between the kinetic energy $T$ and potential energy $V$ +- Note: Dot notation above a variable represents its first derivative. Two dots for the second derivative. + +$$L = T - V$$ + +The integral sums up $L$ along a specific path between $t_1$ and $t_2$ + + + + +# Minkowski Space + +```{python} +#| label: fig-minkowski_light_cone +#| fig-cap: Light Cone in Minkowski Space. A stationary observer moves vertically along a 'World Line' +fig, ax = plt.subplots(figsize=(8, 8)) + +# Plot the light cone (x = ±ct) +t = np.linspace(-5, 5, 100) +ax.plot(t, t, linestyle='--', color='orange', label='Light Cone: $x = ct$') +ax.plot(t, -t, linestyle='--', color='orange') + +# Plot the observer's worldline (stationary at x = 0) +ax.plot([0, 0], [-5, 5], color='blue', lw=2, label='Worldline of Observer') + +# Plot space and time axes +ax.axhline(0, color='black', lw=1) # Space axis +ax.axvline(0, color='black', lw=1) # Time axis + +# Annotate causal regions +ax.text(2.5, 2.5, 'Future', fontsize=12, color='green', ha='center') +ax.text(-2.5, -2.5, 'Past', fontsize=12, color='red', ha='center') +ax.text(3.5, -0.5, 'Elsewhere', fontsize=12, color='gray', ha='center') + +# Set labels and limits +ax.set_xlabel('Space ($x$)') +ax.set_ylabel('Time ($t$)') +ax.set_title('Minkowski Space: Light Cone of an Observer') + +# Set plot limits for symmetry +ax.set_xlim(-5, 5) +ax.set_ylim(-5, 5) + +# Add legend +ax.legend(loc='upper right') + +# Display plot +plt.show() + +``` + +```{python} +#| label: corrected_worldlines_lightcone +#| fig-cap: "Minkowski Space: Worldlines for Stationary and Moving Observers" +import numpy as np +import matplotlib.pyplot as plt +import seaborn as sns + + +# Create figure and axis +fig, ax = plt.subplots(figsize=(8, 8)) + +# Generate time values +t = np.linspace(-5, 5, 100) + +# Plot light cone (x = ±t) +ax.plot(t, t, '--', color='orange', label='Light Cone: $x = ct$') +ax.plot(t, -t, '--', color='orange') + +# Plot stationary observer's worldline (x = 0) +ax.plot([0, 0], [-5, 5], color='blue', lw=2, label='Stationary Observer (v = 0)') + +# Plot worldline for v = 0.3c +v1 = 0.3 # Velocity = 0.3c +x1 = v1 * t # Worldline for v = 0.3c +ax.plot(x1, t, '--', color='blue', lw=2, label='Worldline (v = 0.3c)', alpha=0.75) + +# Plot worldline for v = 0.9c +v3 = 0.6 # Velocity = 0.9c +x3 = v3 * t # Worldline for v = 0.9c +ax.plot(x3, t, '--', color='blue', lw=2, label='Worldline (v = 0.6c)', alpha=0.5) + +# Plot worldline for v = 0.9c +v2 = 0.9 # Velocity = 0.9c +x2 = v2 * t # Worldline for v = 0.9c +ax.plot(x2, t, '--', color='blue', lw=2, label='Worldline (v = 0.9c)', alpha=0.25) + +# Set plot limits, labels, and title +ax.set_xlim(-5, 5) +ax.set_ylim(-5, 5) +ax.set_xlabel('Space ($x$)') +ax.set_ylabel('Time ($t$)') +ax.set_title('Minkowski Space: Worldlines for Different Velocities') + +# Add legend +ax.legend(loc='upper left') + +# Show plot +plt.show() + +``` + +```{python} +#| label: adjusted_mechanics_comparison +#| fig-cap: "Corrected Comparison of Newtonian, Lagrangian, and Hamiltonian Mechanics" +# Parameters for the harmonic oscillator +m = 1.0 # Mass (kg) +k = 1.0 # Spring constant (N/m) +omega = np.sqrt(k / m) # Angular frequency +A = 1.0 # Amplitude of oscillation + +# Hamiltonian mechanics: Phase space dynamics +def hamiltonian_system(t: float, y: np.ndarray) -> np.ndarray: + q, p = y # Position and momentum + dqdt = p / m # dx/dt = p / m + dpdt = -k * q # dp/dt = -kx + return [dqdt, dpdt] + +# Solve Hamiltonian system for phase space trajectory +sol = solve_ivp(hamiltonian_system, [0, 10], [A, 0], t_eval=np.linspace(0, 10, 500)) + +# Create figure and axis for the phase space plot +fig, ax = plt.subplots(figsize=(6, 6)) + +# Plot the phase space trajectory (position vs. momentum) +ax.plot(sol.y[0], sol.y[1], color='red') +ax.set_title('Hamiltonian Mechanics: Phase Space Trajectory') +ax.set_xlabel('Position ($q$)') +ax.set_ylabel('Momentum ($p$)') + +# Mark key points on the trajectory +ax.scatter([0, 0], [1, -1], color='black', zorder=5, label='Max Momentum at $x=0$') +ax.axhline(0, color='black', lw=0.8) # Zero momentum line +ax.axvline(0, color='black', lw=0.8) # Zero position line + +# Add legend +ax.legend(loc='upper right') + +# Display plot +plt.show() + +``` + +# Glossary + +- **Momentum** is the quantity of motion an object has—a measure of how hard it is to stop that object. Mathematically, it's $p = mv$, where $p$ is momentum, $m$ is mass, and $v$ is velocity. \ No newline at end of file From a67f2b4340d8ff8e3e0ce129ae74628e7cdf0857 Mon Sep 17 00:00:00 2001 From: jvivian Date: Tue, 15 Oct 2024 20:40:10 -0700 Subject: [PATCH 02/15] Add some about gravity --- posts/classical-mechanics/index.qmd | 70 +++++++++++++++++++++++++++-- 1 file changed, 67 insertions(+), 3 deletions(-) diff --git a/posts/classical-mechanics/index.qmd b/posts/classical-mechanics/index.qmd index e043406..c07a6b0 100644 --- a/posts/classical-mechanics/index.qmd +++ b/posts/classical-mechanics/index.qmd @@ -89,14 +89,14 @@ ax.legend(loc='upper right') plt.show() ``` -Physics equations are incredible because they describe _universal relationships_ about the world we've found ourselves in. For example, the _momentum_ of an object is described by its mass and velocity: $p = mv$. This relationship applies to _every_ object of mass that can have a velocity. And I think that's pretty neat [^2]. This equation also let's us say momentum is proportional to velocity given constant mass. If you double the velocity of an object, you double its momentum. +Physics equations are incredible because they describe _universal relationships_ about the world we've found ourselves in. For example, the _momentum_ of an object is described by its mass and velocity: $p = mv$. This relationship applies to _every_ object of mass that can have a velocity. And I think that's pretty neat [^2]. This equation also states that momentum is proportional to velocity given constant mass. Doubling the velocity of an object doubles its momentum. [^2]: Neature Walk: https://www.youtube.com/watch?v=Hm3JodBR-vs ## Kinematic Equations > Moving Through Time: Universal Relationships of Motion -These describe universal relationships about an object's motion with constant acceleration—useful when you want to predict where or how fast something will be at a given time. +These equations describe universal relationships about an object's motion with constant acceleration—useful for predicting where or how fast something will be at a given time. $$v = v_0 + at$$ @@ -152,6 +152,70 @@ You can plug this equation into $v = v_0 + at$ and do some math to get: $$v^2 = v_0^2 + 2a(x-x0)$$ ::: +## The Universal Law of Gravitation + +```{python} +#| label: fig-gravitational_force_visualization +#| fig-cap: Force vectors (equal and opposite for each mass) generated as a result of their respective masses and distance. +# Gravitational constant (arbitrary scale for visualization) +G = 1.0 + +# Masses of two objects +m1 = 6.0 # Mass of first object +m2 = 2.0 # Mass of second object + +# Positions of the two masses +x1, y1 = -1, 0 # Position of mass 1 +x2, y2 = 1, -2 # Position of mass 2 + +# Distance between the two masses +r = round(np.sqrt((x2 - x1)**2 + (y2 - y1)**2), 2) + +# Calculate the force magnitude using Newton's law of gravitation +F = G * (m1 * m2) / r**2 + +# Unit vector for the direction of the force +dx, dy = (x2 - x1) / r, (y2 - y1) / r + +# Create the figure and axis +fig, ax = plt.subplots(figsize=(8, 6)) + +# Plot the two masses as circles +ax.plot(x1, y1, 'ro', markersize=m1 * 5, label=f'$m_1$: {m1}') +ax.plot(x2, y2, 'bo', markersize=m2 * 5, label=f'$m_2$: {m2}') + +# Plot force vectors (equal and opposite) +ax.arrow(x1, y1, F * dx, F * dy, head_width=0.1, head_length=0.2, + fc='red', ec='red', label='Gravitational Force on $m_1$') +ax.arrow(x2, y2, -F * dx, -F * dy, head_width=0.1, head_length=0.2, + fc='blue', ec='blue', label='Gravitational Force on $m_2$') + +# Set plot limits +# ax.set_xlim(-3, 3) +# ax.set_ylim(-2, 2) + +# Add labels, title, and legend +ax.set_xlabel('x') +ax.set_ylabel('y') +# ax.set_title(f"Gravitational force vectors - Distance: {r}") +ax.legend(loc='upper right') + +# Display the plot +plt.show() +``` + +Newton's Universal Law of Gravitation describes the generated by two masses. This equation also ties together the inverse square law (decay of intensity of effects such as light, radiation, or gravity). + +$$F = G\frac{m_1m_2}{r^2}$$ + +- $F$ is the gravitational force between two masses. +- $G$ is the gravitational constant. +- $m_1$ and $m_2$ are the masses of the two objects. +- $r$ is the distance between the centers of the two masses. + +Why is $r$ squared? Why not quadrupled or halved? The inverse square relationship emerges as a property of living in three dimensions. The volume of a sphere increases with the square of the radius: $4\pi r^2$. + + # Lagrangian and Hamiltonian Mechanics > Different ways to tell the same (Newtonian) story @@ -159,7 +223,7 @@ Instead of forces and acceleration, Lagrangian and Hamiltonian mechanics focuses ```{python} #| label: fig-corrected_mechanics_comparison -#| fig-cap: "Comparison of Newtonian, Lagrangian, and Hamiltonian Mechanics" +#| fig-cap: "Comparison of Newtonian, Lagrangian, and Hamiltonian Mechanics for a simple harmonic oscillator" from scipy.integrate import solve_ivp # Parameters for the harmonic oscillator m = 1.0 # Mass (kg) From c30532f3dbd724a5295942826ac4c6da88fe1d25 Mon Sep 17 00:00:00 2001 From: jvivian Date: Fri, 1 Nov 2024 00:10:34 -0700 Subject: [PATCH 03/15] Updates --- posts/classical-mechanics/index.qmd | 187 +++++++++++++++++++++------- 1 file changed, 145 insertions(+), 42 deletions(-) diff --git a/posts/classical-mechanics/index.qmd b/posts/classical-mechanics/index.qmd index c07a6b0..3cf3eed 100644 --- a/posts/classical-mechanics/index.qmd +++ b/posts/classical-mechanics/index.qmd @@ -23,9 +23,21 @@ import seaborn as sns plt.style.use('ggplot') ``` -# Acceleration is Absolute +# A Journey Through (Space)Time > _I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me_ [^1] +This overview is roughly structured by the concepts I found most profound while reading and researching topics in Sean Carroll's book _The Biggest Ideas in The Universe: Space, Time, and Motion_ + +- While velocity only has meaning in other reference frames, acceleration is _absolute_, and can be identified intriniscally without one (in most cases). +- The squared term in the inverse square law is derived from the spacial geometry we live in (three dimensions). +- Magnetism only exists as a relativistic effect when an object moves relative to an electric field, which helps explain why we have never observed a magnetic monopole. +- The path a particle or system takes is always the most "economical" route through spacetime (via minimization/maximization of **action**) +- Conservation laws arise directly from symmetries in the laws of physics. I.e. The invariance of a system's laws relate directly to a given conservation law: time invariance $\Leftrightarrow$ energy conservation, spatial invariance $\Leftrightarrow$ momentum conservation, and rotational symmetry $\Leftrightarrow$ angular momentum conservation. +- The speed of light, $c$, provides a conversion factor between time and space. Like energy $\Leftrightarrow$ mass and electricity $\Leftrightarrow$ magnetism, they are aspects of the same phenomena. +- Schwartzchild's solution to General Relativity, contained a mathematical singularity which was originally thought to be a mistake but is now known to be a _real_ singularity in spacetime. + +# Acceleration is Absolute + [^1]: Isaac Newton. "Anecdotes, Observations and Characters, of Books and Men" Vol 1 (page 158). Spence, J. 1820. Imagine yourself in space with no nearby objects, only distant stars; would it be possible to tell you are moving? Nope. As Einstein later proved, velocity only has context relative to another reference frame. Two people moving in in the same environment past each other would not be able to conclude whether one person was stationary and the other was moving or if both people are moving. @@ -215,6 +227,118 @@ $$F = G\frac{m_1m_2}{r^2}$$ Why is $r$ squared? Why not quadrupled or halved? The inverse square relationship emerges as a property of living in three dimensions. The volume of a sphere increases with the square of the radius: $4\pi r^2$. +# Maxwell's Equations + +## **Gauss's Law (Electric Field)** + +$$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$$ + +**Meaning:** +Gauss's Law describes the relationship between electric charges and the electric field they produce. Specifically, it states that the electric flux (the "flow" of the electric field) through a closed surface is proportional to the total charge enclosed within that surface. + +- $\nabla \cdot \mathbf{E}$: This is the divergence of the electric field $\mathbf{E}$, which essentially measures how much the electric field is "spreading out" from a point. +- $\rho$: This represents the **charge density** (charge per unit volume). The more charge you have in a given region, the stronger the electric field diverges from that region. +- $\epsilon_0$: This is the **permittivity of free space**, a constant that determines how electric fields interact in a vacuum. + +**Interpretation:** +Gauss's Law tells us that **electric charges are the source of electric fields**. Positive charges create outward-directed electric fields, and negative charges create inward-directed fields. + +For example, the electric field around a point charge (like an electron) points away from the charge if it's positive, and towards the charge if it's negative. The strength of the field is directly related to the amount of charge. + +## **Gauss's Law for Magnetism** + +$$\nabla \cdot \mathbf{B} = 0$$ + +**Meaning:** +This law states that there are **no magnetic monopoles** — that is, there are no isolated north or south magnetic poles. Magnetic fields ($\mathbf{B}$) always form closed loops. Every magnetic field line that leaves a north pole must eventually return to a south pole. + +- $\nabla \cdot \mathbf{B}$: This is the divergence of the magnetic field $\mathbf{B}$, which measures how much the magnetic field is "spreading out" from a point. +- The fact that it's equal to zero means that the net magnetic "flux" through any closed surface is always zero, implying that magnetic field lines don’t begin or end at any point (unlike electric fields, which start or end on charges). + +**Interpretation:** +This law tells us that there are no magnetic charges (monopoles) in nature — we only observe dipoles (like bar magnets, with both north and south poles). Magnetic field lines form loops, with no starting or ending point. + +## **Faraday's Law of Induction** +$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$ + +**Meaning:** +Faraday’s Law explains how a changing magnetic field creates an electric field. If you have a time-varying magnetic field ($\mathbf{B}$), it induces a circulating electric field ($\mathbf{E}$). This is the principle behind electromagnetic induction, which is used in transformers and electric generators. + +- **$\nabla \times \mathbf{E}$**: This is the curl of the electric field $\mathbf{E}$, which measures how much the electric field is "circulating" around a point. +- **$\frac{\partial \mathbf{B}}{\partial t}$**: This is the time rate of change of the magnetic field. It represents how the magnetic field is changing in time. +- The negative sign indicates that the induced electric field opposes the change in the magnetic field (this is Lenz's law). + +**Interpretation:** +Faraday’s Law describes how **changing magnetic fields induce electric fields**. For example, moving a magnet through a coil of wire induces a current in the wire, because the magnetic field through the coil is changing. + +In a more general sense, this law explains how electric generators work. When a magnet is spun around near a coil of wire, the changing magnetic field induces an electric current in the wire, which we can then use as electrical power. + +## **Ampère's Law (with Maxwell’s Correction)** + +$$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$ + +**Meaning:** +Ampère’s Law relates electric currents and changing electric fields to the magnetic field they produce. It says that magnetic fields can be generated in two ways: +- By electric currents (moving charges), as described by the first term, $\mu_0 \mathbf{J}$. +- By changing electric fields, as described by the second term, $\mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$, which is called the **displacement current** (introduced by Maxwell). + +- $\nabla \times \mathbf{B}$: This is the curl of the magnetic field $\mathbf{B}$, which measures how much the magnetic field "circulates" around a point. +- $\mathbf{J}$: This is the **current density** (current per unit area). +- $\frac{\partial \mathbf{E}}{\partial t}$: This represents the rate of change of the electric field in time. + +**Interpretation:** +Ampère’s Law with Maxwell’s correction tells us that **currents** (moving charges) and **changing electric fields** generate magnetic fields. Without the time-dependent term $\frac{\partial \mathbf{E}}{\partial t}$, this would just be the original Ampère’s Law, which describes how an electric current creates a magnetic field (like in an electromagnet). Maxwell’s addition was crucial because it showed that even in the absence of a physical current, a changing electric field can still produce a magnetic field. + +This law is key to understanding why **electromagnetic waves** (like light) can propagate through empty space. The changing electric field creates a magnetic field, and the changing magnetic field creates an electric field — a self-propagating wave. + +### The Bigger Picture: Electromagnetic Waves +Maxwell’s equations, taken together, explain how electric and magnetic fields interact and propagate. If you combine Faraday's Law and Ampère’s Law, you can derive the **wave equation** for both electric and magnetic fields, showing that disturbances in these fields propagate at the speed of light. + +This is why light, radio waves, X-rays, and all other electromagnetic radiation are essentially **electromagnetic waves**: oscillating electric and magnetic fields that move through space, even in a vacuum. + +- **Faraday’s Law**: A changing magnetic field creates an electric field. +- **Ampère’s Law**: A changing electric field creates a magnetic field. +Together, these two effects allow electromagnetic waves to propagate through space, with the electric and magnetic fields continuously regenerating each other. + +### Summary of Maxwell's Equations: +1. **Gauss’s Law**: Electric charges produce electric fields. +2. **Gauss’s Law for Magnetism**: There are no magnetic monopoles (magnetic fields form closed loops). +3. **Faraday’s Law**: A changing magnetic field induces an electric field. +4. **Ampère’s Law**: Electric currents and changing electric fields produce magnetic fields. + +These equations form the cornerstone of classical electromagnetism and show how electric and magnetic fields are two aspects of a unified force: **electromagnetism**. They govern everything from the behavior of electric circuits to the propagation of light across the universe. + +```{python} +#| label: fig-gauss_law_electric +#| fig-cap: "Gauss's Law for Electricity: Electric Field Lines Around Charges" +# Create a grid for the field +x, y = np.meshgrid(np.linspace(-2, 2, 400), np.linspace(-2, 2, 400)) + +# Define charge positions and magnitudes (positive and negative) +charges = [(1, (0.5, 0)), (-1, (-0.5, 0))] + +# Initialize the electric field vectors +Ex, Ey = np.zeros(x.shape), np.zeros(y.shape) + +# Loop over the charges and calculate the contribution to the field +for charge, (cx, cy) in charges: + r = np.sqrt((x - cx)**2 + (y - cy)**2) + Ex += charge * (x - cx) / r**3 + Ey += charge * (y - cy) / r**3 + +# Plot the electric field lines +plt.figure(figsize=(6, 6)) +plt.streamplot(x, y, Ex, Ey, color=np.sqrt(Ex**2 + Ey**2), linewidth=1) +plt.plot(0.5, 0, 'ro', markersize=10, label='Positive Charge') +plt.plot(-0.5, 0, 'bo', markersize=10, label='Negative Charge') + +# Labels and title +plt.title("Gauss's Law for Electricity") +plt.legend() +plt.show() +``` + + # Lagrangian and Hamiltonian Mechanics > Different ways to tell the same (Newtonian) story @@ -299,7 +423,6 @@ The integral sums up $L$ along a specific path between $t_1$ and $t_2$ - # Minkowski Space ```{python} @@ -392,47 +515,27 @@ plt.show() ``` -```{python} -#| label: adjusted_mechanics_comparison -#| fig-cap: "Corrected Comparison of Newtonian, Lagrangian, and Hamiltonian Mechanics" -# Parameters for the harmonic oscillator -m = 1.0 # Mass (kg) -k = 1.0 # Spring constant (N/m) -omega = np.sqrt(k / m) # Angular frequency -A = 1.0 # Amplitude of oscillation - -# Hamiltonian mechanics: Phase space dynamics -def hamiltonian_system(t: float, y: np.ndarray) -> np.ndarray: - q, p = y # Position and momentum - dqdt = p / m # dx/dt = p / m - dpdt = -k * q # dp/dt = -kx - return [dqdt, dpdt] - -# Solve Hamiltonian system for phase space trajectory -sol = solve_ivp(hamiltonian_system, [0, 10], [A, 0], t_eval=np.linspace(0, 10, 500)) - -# Create figure and axis for the phase space plot -fig, ax = plt.subplots(figsize=(6, 6)) - -# Plot the phase space trajectory (position vs. momentum) -ax.plot(sol.y[0], sol.y[1], color='red') -ax.set_title('Hamiltonian Mechanics: Phase Space Trajectory') -ax.set_xlabel('Position ($q$)') -ax.set_ylabel('Momentum ($p$)') -# Mark key points on the trajectory -ax.scatter([0, 0], [1, -1], color='black', zorder=5, label='Max Momentum at $x=0$') -ax.axhline(0, color='black', lw=0.8) # Zero momentum line -ax.axvline(0, color='black', lw=0.8) # Zero position line - -# Add legend -ax.legend(loc='upper right') - -# Display plot -plt.show() - -``` # Glossary -- **Momentum** is the quantity of motion an object has—a measure of how hard it is to stop that object. Mathematically, it's $p = mv$, where $p$ is momentum, $m$ is mass, and $v$ is velocity. \ No newline at end of file +- **Momentum** is the quantity of motion an object has—a measure of how hard it is to stop that object. Mathematically, it's $p = mv$, where $p$ is momentum, $m$ is mass, and $v$ is velocity. +- **Force** is any interaction that causes an object to change its motion (accelerate or decelerate). Newton's Second Law defines force as $F = ma$, where $F$ is the force, $m$ is mass, and $a$ is acceleration. +- **Energy** is the capacity to do work. It exists in various forms, such as kinetic energy, potential energy, thermal energy, etc. In mechanics, the most common types are: + - **Kinetic Energy** is the energy an object has due to its motion: $K = \frac{1}{2}mv^2$. + - **Potential Energy** is the energy an object has due to its position in a force field, such as gravitational potential energy $U = mgh$ for an object in Earth's gravitational field. +- **Work** is the process of energy transfer when a force is applied to an object and it moves: $W = F \cdot d \cdot \cos(\theta)$, where $F$ is the force, $d$ is the displacement, and $\theta$ is the angle between the force and displacement vectors. +- **Newton's Laws of Motion**: + - **First Law (Inertia):** An object will remain at rest or in uniform motion unless acted upon by an external force. + - **Second Law:** The acceleration of an object is proportional to the net force acting on it and inversely proportional to its mass: $F = ma$. + - **Third Law:** For every action, there is an equal and opposite reaction. +- **Maxwell's Equations** describe how electric and magnetic fields interact and propagate: + - **Gauss's Law:** The electric flux through a closed surface is proportional to the charge enclosed. + - **Gauss's Law for Magnetism:** Magnetic monopoles do not exist; the magnetic flux through a closed surface is zero. + - **Faraday's Law of Induction:** A changing magnetic field induces an electric field. + - **Ampère's Law (with Maxwell's correction):** Magnetic fields can be generated by electric currents and changing electric fields. +- **Lagrangian Mechanics** is a reformulation of classical mechanics based on the principle of least action. The **Lagrangian** $L$ is defined as the difference between kinetic and potential energy: $L = T - U$. It leads to the Euler-Lagrange equations, which describe the dynamics of a system. +- **Hamiltonian Mechanics** is another reformulation of classical mechanics, focused on energy conservation. The **Hamiltonian** $H$ is the total energy of the system (kinetic plus potential energy): $H = T + U$. It uses a set of equations called Hamilton's equations to describe the evolution of a system over time. +- **Minkowski Space** is a four-dimensional space combining three-dimensional space with time into a single manifold, used in the theory of special relativity. It allows for a unified treatment of space and time with the **spacetime interval**: $s^2 = (ct)^2 - x^2 - y^2 - z^2$, where $c$ is the speed of light. Note how $ct$ makes the units consistent. +- **Einstein's Theory of Special Relativity** states that the laws of physics are the same in all inertial frames, and that the speed of light in a vacuum is constant. This leads to the famous equation $E = mc^2$, which relates mass and energy. +- **Einstein's Theory of General Relativity** generalizes special relativity to include gravity. It describes _gravity not as a force_ but as the curvature of spacetime caused by mass and energy. Objects follow geodesics, the shortest paths, in this curved spacetime. \ No newline at end of file From 3a7d2f58814d796d7c8f4d3427b66a08cace0050 Mon Sep 17 00:00:00 2001 From: jvivian Date: Fri, 1 Nov 2024 15:53:07 -0700 Subject: [PATCH 04/15] Update intro --- posts/classical-mechanics/index.qmd | 22 +++++++++++----------- 1 file changed, 11 insertions(+), 11 deletions(-) diff --git a/posts/classical-mechanics/index.qmd b/posts/classical-mechanics/index.qmd index 3cf3eed..3392aba 100644 --- a/posts/classical-mechanics/index.qmd +++ b/posts/classical-mechanics/index.qmd @@ -1,6 +1,6 @@ --- -title: "Pythagorean Time Dilation" -subtitle: "A Summary of Classical Mechanics" +title: "A Journey Through (Space)Time" +subtitle: "Summary of Classical Mechanics" author: John Vivian date: 2024-10-15 format: @@ -23,18 +23,18 @@ import seaborn as sns plt.style.use('ggplot') ``` -# A Journey Through (Space)Time > _I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me_ [^1] -This overview is roughly structured by the concepts I found most profound while reading and researching topics in Sean Carroll's book _The Biggest Ideas in The Universe: Space, Time, and Motion_ +This overview is structured by the concepts I found most profound while reading and researching topics in Sean Carroll's book _The Biggest Ideas in The Universe: Space, Time, and Motion_. + +- While velocity only has meaning relative to other reference frames, acceleration is **absolute**. This means it can be detected intrinsically, even without comparing it to anything else (e.g., feeling pushed back in a car as it accelerates). +- The inverse square law's squared term stems from our three-dimensional geometry. In three-dimensional space, the force from a point source dissipates as it spreads over the surface of a growing sphere. +- Magnetism arises as a relativistic effect due to relative motion between electric charges, explaining why isolated magnetic monopoles haven't been observed. +- The path a particle or system takes is always the most "economical" route through spacetime, achieved by minimizing or maximizing **action** in a system. +- Conservation laws emerge directly from the symmetries inherent in physical laws. For example, time invariance leads to energy conservation, spatial invariance to momentum conservation, and rotational symmetry to angular momentum conservation. +- The speed of light, $c$, provides a conversion factor between time and space, just as energy and mass are related, or electric and magnetic fields are two aspects of the same force. +- Schwarzschild's solution to Einstein's field equations in General Relativity was initially thought incomplete because of a "mathematical singularity." This singularity, however, represents a genuine point of infinite curvature in spacetime—a black hole. -- While velocity only has meaning in other reference frames, acceleration is _absolute_, and can be identified intriniscally without one (in most cases). -- The squared term in the inverse square law is derived from the spacial geometry we live in (three dimensions). -- Magnetism only exists as a relativistic effect when an object moves relative to an electric field, which helps explain why we have never observed a magnetic monopole. -- The path a particle or system takes is always the most "economical" route through spacetime (via minimization/maximization of **action**) -- Conservation laws arise directly from symmetries in the laws of physics. I.e. The invariance of a system's laws relate directly to a given conservation law: time invariance $\Leftrightarrow$ energy conservation, spatial invariance $\Leftrightarrow$ momentum conservation, and rotational symmetry $\Leftrightarrow$ angular momentum conservation. -- The speed of light, $c$, provides a conversion factor between time and space. Like energy $\Leftrightarrow$ mass and electricity $\Leftrightarrow$ magnetism, they are aspects of the same phenomena. -- Schwartzchild's solution to General Relativity, contained a mathematical singularity which was originally thought to be a mistake but is now known to be a _real_ singularity in spacetime. # Acceleration is Absolute From 65b63269dcaaa4a938dc622c2ff1fe7debbba301 Mon Sep 17 00:00:00 2001 From: jvivian Date: Fri, 1 Nov 2024 16:00:42 -0700 Subject: [PATCH 05/15] Update acceleration section --- posts/classical-mechanics/index.qmd | 40 +++++++++++++---------------- 1 file changed, 18 insertions(+), 22 deletions(-) diff --git a/posts/classical-mechanics/index.qmd b/posts/classical-mechanics/index.qmd index 3392aba..4f1e9e4 100644 --- a/posts/classical-mechanics/index.qmd +++ b/posts/classical-mechanics/index.qmd @@ -25,6 +25,8 @@ plt.style.use('ggplot') > _I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me_ [^1] +[^1]: Isaac Newton. "Anecdotes, Observations and Characters, of Books and Men" Vol 1 (page 158). Spence, J. 1820. + This overview is structured by the concepts I found most profound while reading and researching topics in Sean Carroll's book _The Biggest Ideas in The Universe: Space, Time, and Motion_. - While velocity only has meaning relative to other reference frames, acceleration is **absolute**. This means it can be detected intrinsically, even without comparing it to anything else (e.g., feeling pushed back in a car as it accelerates). @@ -38,52 +40,47 @@ This overview is structured by the concepts I found most profound while reading # Acceleration is Absolute -[^1]: Isaac Newton. "Anecdotes, Observations and Characters, of Books and Men" Vol 1 (page 158). Spence, J. 1820. - -Imagine yourself in space with no nearby objects, only distant stars; would it be possible to tell you are moving? Nope. As Einstein later proved, velocity only has context relative to another reference frame. Two people moving in in the same environment past each other would not be able to conclude whether one person was stationary and the other was moving or if both people are moving. - -Oddly, _acceleration_ is not the same. If you were suddenly accelerated while in the space suit you would feel it; you would jerk forward, feel the suit push against your chest, etc. The reason for this is acceleration produces a _force gradient_, as opposed to velocity which does not require a force to keep you moving (an object in motion, yada yada...). +Imagine yourself in deep space, with only distant stars as a backdrop. Would it be possible to tell if you were moving? Nope. As Einstein showed, velocity only has meaning relative to another reference frame. If two people drifted past each other, neither could determine who was stationary or moving, or if they were both moving at once. -You can build your own accelerator-detector just by hanging a weight from a spring. If you begin accelerating left or right, the spring would stretch in the opposite direction. The weight needs to hang to work, because if you turn it sideways the weight will fall towards the ground, even if remain perfectly still. You _are_ accelerating, and have been for quite some time, directly towards the center of the planet. We experience this acceleration as _weight_, which is canceled out by the normal force of the planet pushing back up against us. +Acceleration, however, is a different story. If you suddenly accelerated in your spacesuit, you would feel it: a jerk forward, the suit pressing against your chest, etc. This is because acceleration produces a _force gradient_, unlike velocity, which doesn’t require a force to sustain it (thanks to the principle of inertia). +You could build a simple “accelerometer” by hanging a weight from a spring. If you accelerated left or right, the spring would stretch in the opposite direction. The weight hanging is crucial, though: if you turned it horizontally, the weight would simply fall downward, even while stationary. This is because you _are_ accelerating—toward the center of the Earth. We perceive this continuous acceleration as _weight_, countered by the normal force of the ground pushing up against us. ```{python} -#| label: fig-spring_weight_visualization -#| fig-cap: "Spring with hanging weight, which stretches to the left if accelerated to the right." +#| label: fig-spring_weight_visualization-dffad +#| fig-cap: "Spring with a hanging weight, stretching left as the system accelerates to the right." # Parameters for the spring and weight k = 50 # Spring constant (N/m) mass = 1.0 # Mass of the weight (kg) g = 9.81 # Gravitational acceleration (m/s^2) -equilibrium_length = 2.0 # Length of the spring with no weight (m) +equilibrium_length = 2.0 # Length of the spring at equilibrium (m) -# Calculate the displacement due to the weight -stretch = mass * g / k # Stretch caused by the weight +# Calculate displacement caused by the weight’s force +stretch = mass * g / k # Stretch from the weight’s force on the spring -# Create the spring curve using sine waves for aesthetics +# Define a wavy line for visualizing the spring def spring_curve(x: np.ndarray, stretch: float) -> np.ndarray: - """Creates a wavy curve representing the spring.""" + """Generates a wave pattern for visualizing a stretched spring.""" return np.sin(10 * np.pi * x) * 0.1 - x * stretch -# Generate the spring curve before and after stretching -x = np.linspace(0, 1, 500) # x-axis points along the spring +# Generate spring curves for before and after stretching +x = np.linspace(0, 1, 500) # Position along the spring length y_unstretched = spring_curve(x, 0) # Unstretched spring y_stretched = spring_curve(x, stretch) # Stretched spring # Create the figure and axis fig, ax = plt.subplots(figsize=(6, 8)) -# Plot the unstretched and stretched spring curves +# Plot both the unstretched and stretched spring curves ax.plot(y_unstretched, -x * equilibrium_length, 'k--', alpha=0.5, label='Unstretched Spring') ax.plot(y_stretched, -x * (equilibrium_length + stretch), 'b-', label='Stretched Spring') -# Plot the weight as a black circle +# Represent the weight as a red circle ax.plot(0, -(equilibrium_length + stretch), 'ro', markersize=15, label='Weight') -# Plot the gravitational force vector (downward) +# Plot gravitational and spring force vectors ax.arrow(0, -(equilibrium_length + stretch), 0, -0.5, head_width=0.1, head_length=0.1, fc='blue', ec='blue', label='Gravitational Force $F_g$') - -# Plot the spring force vector (upward) ax.arrow(0, -(equilibrium_length + stretch), 0, 0.5, head_width=0.1, head_length=0.1, fc='green', ec='green', label='Spring Force $F_s$') @@ -92,7 +89,6 @@ ax.set_xlim(-0.5, 0.5) ax.set_ylim(-3.5, 0.5) ax.set_xlabel('Spring Width') ax.set_ylabel('Height') -# ax.set_title('Spring with Hanging Weight Demonstrating Acceleration') # Add legend ax.legend(loc='upper right') @@ -101,7 +97,7 @@ ax.legend(loc='upper right') plt.show() ``` -Physics equations are incredible because they describe _universal relationships_ about the world we've found ourselves in. For example, the _momentum_ of an object is described by its mass and velocity: $p = mv$. This relationship applies to _every_ object of mass that can have a velocity. And I think that's pretty neat [^2]. This equation also states that momentum is proportional to velocity given constant mass. Doubling the velocity of an object doubles its momentum. +Physics equations are extraordinary because they describe _universal relationships_ in the natural world. For instance, the _momentum_ of an object is simply the product of its mass and velocity: $p = mv$. This relationship applies universally to every object with mass and velocity. And I think that's pretty neat [^2]. Notably, the equation shows that momentum is directly proportional to velocity for a constant mass—doubling an object’s velocity doubles its momentum. [^2]: Neature Walk: https://www.youtube.com/watch?v=Hm3JodBR-vs From 708c376e2fb44162e79905520e1837902d957e6a Mon Sep 17 00:00:00 2001 From: John Vivian Date: Sat, 2 Nov 2024 21:44:54 -0700 Subject: [PATCH 06/15] Huge update to section on Maxwell's Equations --- posts/classical-mechanics/index.qmd | 261 +++++++++++++++++++++------- 1 file changed, 203 insertions(+), 58 deletions(-) diff --git a/posts/classical-mechanics/index.qmd b/posts/classical-mechanics/index.qmd index 3cf3eed..6b73f0e 100644 --- a/posts/classical-mechanics/index.qmd +++ b/posts/classical-mechanics/index.qmd @@ -10,8 +10,8 @@ format: toc-depth: 2 echo: false editor: visual -execute: - render-on-save: true +execute: + render-on-save: true tools: - mermaid --- @@ -32,9 +32,9 @@ This overview is roughly structured by the concepts I found most profound while - The squared term in the inverse square law is derived from the spacial geometry we live in (three dimensions). - Magnetism only exists as a relativistic effect when an object moves relative to an electric field, which helps explain why we have never observed a magnetic monopole. - The path a particle or system takes is always the most "economical" route through spacetime (via minimization/maximization of **action**) -- Conservation laws arise directly from symmetries in the laws of physics. I.e. The invariance of a system's laws relate directly to a given conservation law: time invariance $\Leftrightarrow$ energy conservation, spatial invariance $\Leftrightarrow$ momentum conservation, and rotational symmetry $\Leftrightarrow$ angular momentum conservation. +- Conservation laws arise directly from symmetries in the laws of physics. I.e. The invariance of a system's laws relate directly to a given conservation law: time invariance $\rightarrow$ energy conservation, spatial invariance $\rightarrow$ momentum conservation, and rotational symmetry $\rightarrow$ angular momentum conservation. - The speed of light, $c$, provides a conversion factor between time and space. Like energy $\Leftrightarrow$ mass and electricity $\Leftrightarrow$ magnetism, they are aspects of the same phenomena. -- Schwartzchild's solution to General Relativity, contained a mathematical singularity which was originally thought to be a mistake but is now known to be a _real_ singularity in spacetime. +- Schwartzchild's solution to General Relativity contained a mathematical singularity -- originally thought to be a _coordinate singularity_ -- turned out to be a real _curvature singularity_ in spacetime: a blackhole. # Acceleration is Absolute @@ -228,56 +228,205 @@ $$F = G\frac{m_1m_2}{r^2}$$ Why is $r$ squared? Why not quadrupled or halved? The inverse square relationship emerges as a property of living in three dimensions. The volume of a sphere increases with the square of the radius: $4\pi r^2$. # Maxwell's Equations +> Quote here + +Maxwell's Equations elegantly describe the behavior of electric and magnetic fields, revealing that these two forces are not separate but deeply interconnected. They provide a unified framework for understanding how electric charges and currents create fields that interact, propagate, and influence each other, ultimately giving rise to light itself. ## **Gauss's Law (Electric Field)** $$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$$ -**Meaning:** -Gauss's Law describes the relationship between electric charges and the electric field they produce. Specifically, it states that the electric flux (the "flow" of the electric field) through a closed surface is proportional to the total charge enclosed within that surface. +:::{.callout-note collapse="true"} +## Understanding the Del Operator ($\nabla$) in Maxwell’s Equations + +The **del operator** ($\nabla$) is a vector operator that captures how quantities change in space, and it’s central to expressing Maxwell’s equations in a compact, powerful form. + +1. **Gradient ($\nabla \phi$)**: When applied to a scalar field $\phi$, $\nabla$ (del) gives the **gradient**, pointing in the direction of the greatest rate of change. For example, in an electric potential field $\phi$, the gradient $\vec{E} = -\nabla\phi$ tells us how the electric field varies in space. + +2. **Divergence ($\nabla \cdot$)**: When $\nabla$ is applied to a vector field with a dot product ($\nabla \cdot$), it gives the **divergence**. Divergence measures the “outflow” or “inflow” of a field at a point: + - **Gauss’s Law for Electricity** $(\nabla·\vec{E} = \frac{ρ}{ε_0})$: This tells us that the electric field $\vec{E}$ diverges from electric charges. + - **Gauss’s Law for Magnetism** $(\nabla·\vec{B} = 0)$: This shows that magnetic field lines have no divergence—they form closed loops, indicating no magnetic monopoles. + +3. **Curl ($\nabla \times$)**: When $\nabla$ is applied to a vector field with a cross product ($\nabla \times$), it gives the **curl**, measuring the “rotation” or “circulation” of a field: + - **Faraday’s Law** $(\nabla×\vec{E} = -\frac{∂\vec{B}}{∂t})$: A changing magnetic field $\vec{B}$ induces a circulating electric field $\vec{E}$. + - **Ampère’s Law (with Maxwell’s Correction)** $(\nabla×\vec{B} = μ_0\vec{J} + μ_0ε_0\frac{∂\vec{E}}{∂t})$: A changing electric field $\vec{E}$ or current density $\vec{J}$ induces a circulating magnetic field $\vec{B}$. + +### Summary: +The del operator ($\nabla$) is a vector operator that operates on a: +- **Gradient ($\nabla\phi$)**: scalar field, producing a vector field. +- **Divergence ($\nabla\cdot$)**: vector field, producing a scalar field. +- **Curl ($\nabla\times$)**: vector field, producing a vector field. +::: + + +Gauss's Law describes the relationship between electric charges and the electric field they produce. Specifically, it states that the electric flux (the "flow" of the electric field) through a closed surface is proportional to the total charge enclosed within that surface. For example, the electric field around a point charge (like an electron) points away from the charge if it's positive, and towards the charge if it's negative. The strength of the field is directly related to the amount of charge. - $\nabla \cdot \mathbf{E}$: This is the divergence of the electric field $\mathbf{E}$, which essentially measures how much the electric field is "spreading out" from a point. - $\rho$: This represents the **charge density** (charge per unit volume). The more charge you have in a given region, the stronger the electric field diverges from that region. - $\epsilon_0$: This is the **permittivity of free space**, a constant that determines how electric fields interact in a vacuum. -**Interpretation:** -Gauss's Law tells us that **electric charges are the source of electric fields**. Positive charges create outward-directed electric fields, and negative charges create inward-directed fields. +```{python} +#| label: fig-gauss_law_electric +#| fig-cap: "Gauss's Law for Electricity: Electric Field Lines Around Charges" +# Create a grid for the field +x, y = np.meshgrid(np.linspace(-2, 2, 400), np.linspace(-2, 2, 400)) + +# Define charge positions and magnitudes (positive and negative) +charges = [(1, (0.5, 0)), (-1, (-0.5, 0))] + +# Initialize the electric field vectors +Ex, Ey = np.zeros(x.shape), np.zeros(y.shape) + +# Loop over the charges and calculate the contribution to the field +for charge, (cx, cy) in charges: + r = np.sqrt((x - cx)**2 + (y - cy)**2) + Ex += charge * (x - cx) / r**3 + Ey += charge * (y - cy) / r**3 + +# Plot the electric field lines +plt.figure(figsize=(6, 6)) +plt.streamplot(x, y, Ex, Ey, color=np.sqrt(Ex**2 + Ey**2), linewidth=1) +plt.plot(0.5, 0, 'ro', markersize=10, label='Positive Charge') +plt.plot(-0.5, 0, 'bo', markersize=10, label='Negative Charge') + +# Labels and title +plt.title("Gauss's Law for Electricity") +plt.legend() +plt.show() +``` -For example, the electric field around a point charge (like an electron) points away from the charge if it's positive, and towards the charge if it's negative. The strength of the field is directly related to the amount of charge. ## **Gauss's Law for Magnetism** $$\nabla \cdot \mathbf{B} = 0$$ -**Meaning:** -This law states that there are **no magnetic monopoles** — that is, there are no isolated north or south magnetic poles. Magnetic fields ($\mathbf{B}$) always form closed loops. Every magnetic field line that leaves a north pole must eventually return to a south pole. +This law states that there are **no magnetic monopoles** — that is, no isolated north or south magnetic poles. Magnetic fields ($\mathbf{B}$) always form closed loops. Every magnetic field line that leaves a north pole must eventually return to a south pole. - $\nabla \cdot \mathbf{B}$: This is the divergence of the magnetic field $\mathbf{B}$, which measures how much the magnetic field is "spreading out" from a point. - The fact that it's equal to zero means that the net magnetic "flux" through any closed surface is always zero, implying that magnetic field lines don’t begin or end at any point (unlike electric fields, which start or end on charges). -**Interpretation:** -This law tells us that there are no magnetic charges (monopoles) in nature — we only observe dipoles (like bar magnets, with both north and south poles). Magnetic field lines form loops, with no starting or ending point. +```{python} +#| label: fig-magnetic +#| fig-cap: Magnetic field lines around a magnetic point + +# Grid for the contour plot +x = np.linspace(-3, 3, 100) +y = np.linspace(-3, 3, 100) +X, Y = np.meshgrid(x, y) + +# Calculate a hypothetical magnetic dipole field in 2D for visualization +# Assuming a simple dipole approximation +B_x = -Y / (X**2 + Y**2)**1.5 # Field component in x direction +B_y = X / (X**2 + Y**2)**1.5 # Field component in y direction + +# Magnitude of the field for contour levels +B_magnitude = np.sqrt(B_x**2 + B_y**2) + +# Create the contour plot +fig, ax = plt.subplots(figsize=(8, 8)) + +# Plot the field lines as contours +contour = ax.contour(X, Y, B_magnitude, levels=10, cmap="viridis", alpha=0.75) +ax.clabel(contour, inline=True, fontsize=8, fmt="%.2f") + +# Add streamplot to show the direction of the magnetic field +ax.streamplot(X, Y, B_x, B_y, color="darkred", linewidth=1, density=1.5, arrowsize=1.5) + +# Plot Gaussian surface as a dashed circle to emphasize the zero net flux +surface_radius = 2.5 +theta = np.linspace(0, 2 * np.pi, 100) +x_surface = surface_radius * np.cos(theta) +y_surface = surface_radius * np.sin(theta) +ax.plot(x_surface, y_surface, 'k--', label="Gaussian Surface") + +# Annotate the North and South poles for reference +ax.text(0, 2.8, "N", ha='center', va='center', fontsize=14, color='blue') +ax.text(0, -2.8, "S", ha='center', va='center', fontsize=14, color='blue') + +# Labels, legend, and aesthetic adjustments +ax.legend(loc="upper right") +ax.axis("equal") +ax.axis("off") + +# Display the plot +plt.show() +``` + ## **Faraday's Law of Induction** $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$ **Meaning:** -Faraday’s Law explains how a changing magnetic field creates an electric field. If you have a time-varying magnetic field ($\mathbf{B}$), it induces a circulating electric field ($\mathbf{E}$). This is the principle behind electromagnetic induction, which is used in transformers and electric generators. +Faraday’s Law explains how a changing magnetic field creates an electric field. If you have a time-varying magnetic field ($\mathbf{B}$), it induces a circulating electric field ($\mathbf{E}$). This is the principle behind electromagnetic induction, which is used in transformers and electric generators. When a magnet is spun around near a coil of wire, the changing magnetic field induces an electric current in the wire, which we can then use as electrical power. - **$\nabla \times \mathbf{E}$**: This is the curl of the electric field $\mathbf{E}$, which measures how much the electric field is "circulating" around a point. - **$\frac{\partial \mathbf{B}}{\partial t}$**: This is the time rate of change of the magnetic field. It represents how the magnetic field is changing in time. - The negative sign indicates that the induced electric field opposes the change in the magnetic field (this is Lenz's law). -**Interpretation:** -Faraday’s Law describes how **changing magnetic fields induce electric fields**. For example, moving a magnet through a coil of wire induces a current in the wire, because the magnetic field through the coil is changing. +```{python} +#| label: fig-faraday +#| fig-cap: Changing magnetic fields induce electric fields + +import numpy as np +import matplotlib.pyplot as plt +from mpl_toolkits.mplot3d import Axes3D + +# Set up the figure and 3D axes with ggplot style +plt.style.use('ggplot') +fig = plt.figure(figsize=(10, 6)) +ax = fig.add_subplot(111, projection='3d') + +# Parameters for the waves +x = np.linspace(0, 20, 1000) # Position along the propagation direction (x-axis) +k = 2 * np.pi / 5 # Wave number +omega = 2 * np.pi / 5 # Angular frequency +t = 0 # Time (static snapshot) +E0 = 1 # Electric field amplitude +B0 = 1 # Magnetic field amplitude + +# Calculate the electric and magnetic fields as sinusoidal waves +E = E0 * np.sin(k * x - omega * t) # Electric field (along y-axis) +B = B0 * np.sin(k * x - omega * t) # Magnetic field (along z-axis) + +# Plot the electric field +ax.plot(x, E, np.zeros_like(x), color='blue', label='Electric Field (E)', linewidth=2) +# Plot the magnetic field +ax.plot(x, np.zeros_like(x), B, color='red', label='Magnetic Field (B)', linewidth=2) + +# Add a dotted arrow along the x-axis to represent the direction of wave propagation +ax.plot(x, np.zeros_like(x), np.zeros_like(x), 'k--', label='Propagation Direction (x)', linewidth=1.5) + +# Customize the view angle for a clear perspective +ax.view_init(elev=30, azim=60) + +# Labels and legend for clarity +ax.set_xlabel('Propagation Direction (x)', fontsize=12) +ax.set_ylabel('Electric Field (E)', fontsize=12) +ax.set_zlabel('Magnetic Field (B)', fontsize=12) +ax.legend(fontsize=12) + +# Minimalist aesthetics: turn off grid panes for a cleaner look +ax.grid(False) +ax.xaxis.pane.fill = False +ax.yaxis.pane.fill = False +ax.zaxis.pane.fill = False +ax.xaxis.pane.set_edgecolor('w') +ax.yaxis.pane.set_edgecolor('w') +ax.zaxis.pane.set_edgecolor('w') +ax.set_xticks([]) +ax.set_yticks([]) +ax.set_zticks([]) + +# Show the plot +plt.tight_layout() +plt.show() +``` -In a more general sense, this law explains how electric generators work. When a magnet is spun around near a coil of wire, the changing magnetic field induces an electric current in the wire, which we can then use as electrical power. ## **Ampère's Law (with Maxwell’s Correction)** $$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$ -**Meaning:** Ampère’s Law relates electric currents and changing electric fields to the magnetic field they produce. It says that magnetic fields can be generated in two ways: - By electric currents (moving charges), as described by the first term, $\mu_0 \mathbf{J}$. - By changing electric fields, as described by the second term, $\mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$, which is called the **displacement current** (introduced by Maxwell). @@ -286,58 +435,54 @@ Ampère’s Law relates electric currents and changing electric fields to the ma - $\mathbf{J}$: This is the **current density** (current per unit area). - $\frac{\partial \mathbf{E}}{\partial t}$: This represents the rate of change of the electric field in time. -**Interpretation:** Ampère’s Law with Maxwell’s correction tells us that **currents** (moving charges) and **changing electric fields** generate magnetic fields. Without the time-dependent term $\frac{\partial \mathbf{E}}{\partial t}$, this would just be the original Ampère’s Law, which describes how an electric current creates a magnetic field (like in an electromagnet). Maxwell’s addition was crucial because it showed that even in the absence of a physical current, a changing electric field can still produce a magnetic field. -This law is key to understanding why **electromagnetic waves** (like light) can propagate through empty space. The changing electric field creates a magnetic field, and the changing magnetic field creates an electric field — a self-propagating wave. - -### The Bigger Picture: Electromagnetic Waves -Maxwell’s equations, taken together, explain how electric and magnetic fields interact and propagate. If you combine Faraday's Law and Ampère’s Law, you can derive the **wave equation** for both electric and magnetic fields, showing that disturbances in these fields propagate at the speed of light. - -This is why light, radio waves, X-rays, and all other electromagnetic radiation are essentially **electromagnetic waves**: oscillating electric and magnetic fields that move through space, even in a vacuum. - -- **Faraday’s Law**: A changing magnetic field creates an electric field. -- **Ampère’s Law**: A changing electric field creates a magnetic field. -Together, these two effects allow electromagnetic waves to propagate through space, with the electric and magnetic fields continuously regenerating each other. +```{python} +#| label: fig-ampere +#| fig-cap: The opposite of Faraday's Law - a time-varying electric field is the source of an induced magnetic field. +# Re-plotting with only "dE/dt" to emphasize the time-varying electric field as the source of the induced magnetic field -### Summary of Maxwell's Equations: -1. **Gauss’s Law**: Electric charges produce electric fields. -2. **Gauss’s Law for Magnetism**: There are no magnetic monopoles (magnetic fields form closed loops). -3. **Faraday’s Law**: A changing magnetic field induces an electric field. -4. **Ampère’s Law**: Electric currents and changing electric fields produce magnetic fields. +# Set up the figure with ggplot style for aesthetics +fig, ax = plt.subplots(figsize=(8, 8)) +ax.set_aspect('equal') -These equations form the cornerstone of classical electromagnetism and show how electric and magnetic fields are two aspects of a unified force: **electromagnetism**. They govern everything from the behavior of electric circuits to the propagation of light across the universe. +# Define parameters for the capacitor plates and electric field lines +plate_width = 1.0 +plate_length = 4.0 +plate_offset = 1.5 -```{python} -#| label: fig-gauss_law_electric -#| fig-cap: "Gauss's Law for Electricity: Electric Field Lines Around Charges" -# Create a grid for the field -x, y = np.meshgrid(np.linspace(-2, 2, 400), np.linspace(-2, 2, 400)) +# Plot the capacitor plates with color-coded charges +ax.plot([-plate_length / 2, plate_length / 2], [plate_offset, plate_offset], color='red', lw=4, label="Capacitor Plate (+)") +ax.plot([-plate_length / 2, plate_length / 2], [-plate_offset, -plate_offset], color='black', lw=4, label="Capacitor Plate (-)") -# Define charge positions and magnitudes (positive and negative) -charges = [(1, (0.5, 0)), (-1, (-0.5, 0))] +# Electric field lines between capacitor plates (vertical lines indicating field change) +for i in np.linspace(-plate_length / 2, plate_length / 2, 6): + ax.arrow(i, -plate_offset, 0, 2 * plate_offset, head_width=0.1, head_length=0.2, fc='blue', ec='blue') -# Initialize the electric field vectors -Ex, Ey = np.zeros(x.shape), np.zeros(y.shape) +# Magnetic field lines (circling the changing electric field in the gap) +theta = np.linspace(0, 2 * np.pi, 100) +for radius in [2.2, 2.7, 3.2]: # Multiple circles for the magnetic field lines + x = radius * np.cos(theta) + y = radius * np.sin(theta) + ax.plot(x, y, color='orange', linestyle='--', lw=1.5, alpha=0.8, label="Induced Magnetic Field" if radius == 2.2 else "") -# Loop over the charges and calculate the contribution to the field -for charge, (cx, cy) in charges: - r = np.sqrt((x - cx)**2 + (y - cy)**2) - Ex += charge * (x - cx) / r**3 - Ey += charge * (y - cy) / r**3 +# Annotation to highlight the changing electric field (dE/dt) +ax.text(0, -plate_offset - 0.5, "dE/dt (Changing Electric Field)", ha='center', color='blue', fontsize=12) +ax.text(3.5, 0, "B (Induced Magnetic Field)", color='orange', fontsize=12, rotation=90) -# Plot the electric field lines -plt.figure(figsize=(6, 6)) -plt.streamplot(x, y, Ex, Ey, color=np.sqrt(Ex**2 + Ey**2), linewidth=1) -plt.plot(0.5, 0, 'ro', markersize=10, label='Positive Charge') -plt.plot(-0.5, 0, 'bo', markersize=10, label='Negative Charge') +# Minimalist aesthetics +ax.axis("off") +ax.legend(loc="upper right") -# Labels and title -plt.title("Gauss's Law for Electricity") -plt.legend() +# Display the plot +plt.tight_layout() plt.show() ``` +### The Bigger Picture: Electromagnetic Waves +Maxwell’s equations, taken together, explain how electric and magnetic fields interact and propagate. If you combine Faraday's Law and Ampère’s Law, you can derive the **wave equation** for both electric and magnetic fields, showing that disturbances in these fields propagate at the speed of light. Together, these two effects allow electromagnetic waves to propagate through space, with the electric and magnetic fields continuously regenerating each other. + +This is why light, radio waves, X-rays, and all other electromagnetic radiation are essentially **electromagnetic waves**: oscillating electric and magnetic fields that move through space, even in a vacuum. # Lagrangian and Hamiltonian Mechanics @@ -538,4 +683,4 @@ plt.show() - **Hamiltonian Mechanics** is another reformulation of classical mechanics, focused on energy conservation. The **Hamiltonian** $H$ is the total energy of the system (kinetic plus potential energy): $H = T + U$. It uses a set of equations called Hamilton's equations to describe the evolution of a system over time. - **Minkowski Space** is a four-dimensional space combining three-dimensional space with time into a single manifold, used in the theory of special relativity. It allows for a unified treatment of space and time with the **spacetime interval**: $s^2 = (ct)^2 - x^2 - y^2 - z^2$, where $c$ is the speed of light. Note how $ct$ makes the units consistent. - **Einstein's Theory of Special Relativity** states that the laws of physics are the same in all inertial frames, and that the speed of light in a vacuum is constant. This leads to the famous equation $E = mc^2$, which relates mass and energy. -- **Einstein's Theory of General Relativity** generalizes special relativity to include gravity. It describes _gravity not as a force_ but as the curvature of spacetime caused by mass and energy. Objects follow geodesics, the shortest paths, in this curved spacetime. \ No newline at end of file +- **Einstein's Theory of General Relativity** generalizes special relativity to include gravity. It describes _gravity not as a force_ but as the curvature of spacetime caused by mass and energy. Objects follow geodesics, the shortest paths, in this curved spacetime. From 0a0efd27e044719a1cfa6f03b73ae93a06f1ac3c Mon Sep 17 00:00:00 2001 From: John Vivian Date: Sat, 2 Nov 2024 23:47:08 -0700 Subject: [PATCH 07/15] Update Lagrangian / Hamiltonian section --- posts/classical-mechanics/index.qmd | 162 ++++++++++++++++++++++++++-- 1 file changed, 154 insertions(+), 8 deletions(-) diff --git a/posts/classical-mechanics/index.qmd b/posts/classical-mechanics/index.qmd index 6b73f0e..a79be24 100644 --- a/posts/classical-mechanics/index.qmd +++ b/posts/classical-mechanics/index.qmd @@ -26,16 +26,19 @@ plt.style.use('ggplot') # A Journey Through (Space)Time > _I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me_ [^1] -This overview is roughly structured by the concepts I found most profound while reading and researching topics in Sean Carroll's book _The Biggest Ideas in The Universe: Space, Time, and Motion_ +This overview is roughly structured by the concepts I found most profound while reading and researching topics in Sean Carroll's book _The Biggest Ideas in The Universe: Space, Time, and Motion_. - While velocity only has meaning in other reference frames, acceleration is _absolute_, and can be identified intriniscally without one (in most cases). - The squared term in the inverse square law is derived from the spacial geometry we live in (three dimensions). - Magnetism only exists as a relativistic effect when an object moves relative to an electric field, which helps explain why we have never observed a magnetic monopole. -- The path a particle or system takes is always the most "economical" route through spacetime (via minimization/maximization of **action**) +- The path a particle or system takes is always the most "economical" route through spacetime via minimization/maximization of **action**---a scalar quantity that describes how the balance of kinetic versus potential energy of a physical system changes. - Conservation laws arise directly from symmetries in the laws of physics. I.e. The invariance of a system's laws relate directly to a given conservation law: time invariance $\rightarrow$ energy conservation, spatial invariance $\rightarrow$ momentum conservation, and rotational symmetry $\rightarrow$ angular momentum conservation. - The speed of light, $c$, provides a conversion factor between time and space. Like energy $\Leftrightarrow$ mass and electricity $\Leftrightarrow$ magnetism, they are aspects of the same phenomena. +- Lorentz transformations, which describe effects like length contraction and time dilation, can be derived from the invariant spacetime interval---analagous to the Pythagorean theorem---in the geometry of Minkowski space. - Schwartzchild's solution to General Relativity contained a mathematical singularity -- originally thought to be a _coordinate singularity_ -- turned out to be a real _curvature singularity_ in spacetime: a blackhole. +**Disclaimer**: I’m not a mathematician or physicist, so aim to clarify the meaning of each variable and operator in the equations introduced. + # Acceleration is Absolute [^1]: Isaac Newton. "Anecdotes, Observations and Characters, of Books and Men" Vol 1 (page 158). Spence, J. 1820. @@ -427,14 +430,15 @@ plt.show() $$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$ -Ampère’s Law relates electric currents and changing electric fields to the magnetic field they produce. It says that magnetic fields can be generated in two ways: -- By electric currents (moving charges), as described by the first term, $\mu_0 \mathbf{J}$. -- By changing electric fields, as described by the second term, $\mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$, which is called the **displacement current** (introduced by Maxwell). - - $\nabla \times \mathbf{B}$: This is the curl of the magnetic field $\mathbf{B}$, which measures how much the magnetic field "circulates" around a point. - $\mathbf{J}$: This is the **current density** (current per unit area). - $\frac{\partial \mathbf{E}}{\partial t}$: This represents the rate of change of the electric field in time. +Ampère’s Law relates electric currents and changing electric fields to the magnetic field they produce. It says that magnetic fields can be generated in two ways: + +- By electric currents (moving charges), as described by the first term, $\mu_0 \mathbf{J}$. +- By changing electric fields, as described by the second term, $\mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$, which is called the **displacement current** (introduced by Maxwell). + Ampère’s Law with Maxwell’s correction tells us that **currents** (moving charges) and **changing electric fields** generate magnetic fields. Without the time-dependent term $\frac{\partial \mathbf{E}}{\partial t}$, this would just be the original Ampère’s Law, which describes how an electric current creates a magnetic field (like in an electromagnet). Maxwell’s addition was crucial because it showed that even in the absence of a physical current, a changing electric field can still produce a magnetic field. ```{python} @@ -488,7 +492,69 @@ This is why light, radio waves, X-rays, and all other electromagnetic radiation # Lagrangian and Hamiltonian Mechanics > Different ways to tell the same (Newtonian) story -Instead of forces and acceleration, Lagrangian and Hamiltonian mechanics focuses on _paths, energy, and phase space_. The last is particularly useful for understanding field theory. +In Newtonian mechanics, where $F = ma$, we focus on forces acting at a specific point in space and time to answer "What happens at the next $t$?" This approach helps us understand motion in terms of **forces** and **accelerations**. + +In contrast, mechanics based on action principles---Lagrangian and Hamiltonian mechanics---start from a different perspective, emphasizing the concept of **action**. Action represents an energy tradeoff between kinetic and potential energy, shaped by the system's physical properties. Instead of predicting the immediate "next step," these approaches answer questions about _paths_, e.g. which trajectory will efficiently guide a rocket to the moon? + +In lieu of forces, Lagrangian and Hamiltonian mechanics prioritize **paths, energy, and phase space**. Phase space, in particular, provides a powerful framework for visualizing the state of a system and plays a central role in fields like quantum mechanics and field theory. + +To see these differences in action, we can visualize a classical physical system---the **simple harmonic oscillator**---through the lenses of Newtonian, Lagrangian, and Hamiltonian mechanics. + +::: {.callout-note collapse="true"} +## Simple Harmonic Oscillator + +The **Simple Harmonic Oscillator** is a fundamental model of periodic motion, where an object experiences a restoring force proportional to its distance from an equilibrium position. This force always acts toward the equilibrium, driving the object back whenever it is displaced. + +```{python} +#| label: fig-SHO +#| fig-cap: This mass-spring system illustrates the concept of a simple harmonic oscillator, where the restoring force is proportional to displacement. This same principle applies in other contexts such as pendulums, which exhibit harmonic motion at small angles. LC circuits, oscillate energy between electric and magnetic fields. This even extends to quantum harmonic oscillator models which models particles in a potential well. + +fig, ax = plt.subplots(figsize=(8, 4)) + +# Parameters for the mass-spring system visualization +spring_length = 1.0 # length of the spring at rest +displacement = 0.5 # displacement from equilibrium +mass_position = spring_length + displacement # position of the mass due to displacement + +# Draw the spring as a sinusoidal curve +spring_x = np.linspace(0, spring_length + displacement, 100) +spring_y = 0.1 * np.sin(10 * np.pi * spring_x / (spring_length + displacement)) # Wavy pattern for spring effect +ax.plot(spring_x, spring_y, color='black', linewidth=2, label="Spring") + +# Draw the mass as a box +mass_width, mass_height = 0.2, 0.2 +mass_x = [mass_position - mass_width / 2, mass_position + mass_width / 2, + mass_position + mass_width / 2, mass_position - mass_width / 2, + mass_position - mass_width / 2] +mass_y = [-mass_height / 2, -mass_height / 2, mass_height / 2, mass_height / 2, -mass_height / 2] +ax.fill(mass_x, mass_y, color='gray', label="Mass") + +# Indicate equilibrium position with a dashed line +ax.axvline(spring_length, color='blue', linestyle='--', linewidth=1) +ax.text(spring_length, 0.25, 'Equilibrium Position', color='blue', ha='center', fontsize=10) + +# Draw restoring force arrow +ax.arrow(mass_position, 0, -0.3, 0, head_width=0.05, head_length=0.1, fc='red', ec='red', label="Restoring Force") +ax.text(mass_position - 0.35, 0.05, r"$F = -kx$", color='red', fontsize=12) + +# Annotate displacement +ax.annotate("Displacement (x)", xy=(mass_position - displacement / 2, 0.2), xytext=(mass_position + 0.2, 0.3), + arrowprops=dict(facecolor='black', arrowstyle="->"), fontsize=10) + +# Axis limits and labels +ax.set_xlim(0, spring_length + displacement + 0.5) +ax.set_ylim(-0.5, 0.5) +ax.axis("off") # Turn off axis for a clean look + +# Legend and Display +plt.legend(loc="upper left", fontsize=10) + +# Show the plot +plt.show() +``` + +::: + ```{python} #| label: fig-corrected_mechanics_comparison @@ -564,11 +630,91 @@ $$S[q(t)] = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt$$ $$L = T - V$$ -The integral sums up $L$ along a specific path between $t_1$ and $t_2$ +In essence, the action integral sums up the Lagrangian $L$ over a specific path from $t_1$ to $t_2$. This principle of minimizing the action is known as the _principle of least action_, and it gives us the most efficient or "natural" path a system will take. + +### The Euler-Lagrange Equation + +We're still missing one component---a condition that allows us to minimize the Lagrangian. Thankfully someone way smarter than any of us worked out the rather elegant solution, the **Euler-Lagrange equation**: +$$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} = 0$$ + +- **$\frac{\partial L}{\partial \dot{q}}$**: This is the **partial derivative of the Lagrangian with respect to the velocity** $\dot{q}$. It measures how sensitive the Lagrangian is to changes in the system's velocity. +- **$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right)$**: This term takes the time derivative of the above quantity, capturing how this sensitivity changes as the system evolves over time. +- **$\frac{\partial L}{\partial q}$**: This is the **partial derivative of the Lagrangian with respect to position** $q$. It measures how sensitive the Lagrangian is to changes in the system's position. + +The **Euler-Lagrange equation** tells us that, for the actual path taken by the system, these two rates of change are equal and opposite. + +#### Intuition Behind the Equation + +The Euler-Lagrange equation is the mathematical expression of the **principle of least action**. It provides the conditions under which the action $S$ is minimized. + +To build intuition around this concept, let's imagine a ball rolling down a _frictionless_ hill. + +When the ball starts at the top of the hill, it has maximum potential energy (due to height and the acceleration from gravity) and zero kinetic energy (energy due to motion). As it begins rolling down, gravity pulls it lower, decreasing its potential energy. But energy can’t just disappear—instead, it converts into kinetic energy, causing the ball to speed up. + +The ball follows a path where potential energy smoothly turns into kinetic energy as it descends. The principle of least action explains this smooth exchange by "favoring" paths where energy changes are gradual and balanced. + +When the ball reaches the bottom, it’s out of hill to roll down. Its potential energy is minimized, and all the energy it gained from descending has been converted into kinetic energy. + +This trade-off between potential and kinetic energy is natural because it minimizes abrupt changes in energy. A path where energy flowed in sharp bursts or stopped suddenly would require additional forces or disruptions. A smooth path minimizes these disturbances, making it the most efficient in terms of conserving energy while following the laws of physics. + +If this process sounds similar to **heat diffusion**, you're doin good baby. Both are examples of the Universe's tendency towards _equilibrium_ while maintaining _conservation_. + +::: {.callout-tip} +## TLDR +Any change in kinetic energy is exactly offset by an opposite change in potential energy, and vice versa. +::: + + +::: {.callout-note collapse="true"} +## The Action of Simple Harmonic Motion + +To see the action integral in action (hehe), let’s examine a **simple harmonic oscillator**, such as a mass attached to a spring and solve for the Lagrangian. + +**Kinetic Energy $T$** + +For an object of mass $m$ moving with velocity $\dot{q}$, the kinetic energy $T$ is given by: +$$T = \frac{1}{2} m \dot{q}^2$$ + +**Potential Energy $V$** + +For a spring with spring constant $k$, the potential energy $V$ depends on the displacement $q$ from equilibrium: +$$V = \frac{1}{2} k q^2$$ + +**Lagrangian $L = T - V$** + +Substituting the expressions for $T$ and $V$: + +$$L = T - V = \frac{1}{2} m \dot{q}^2 - \frac{1}{2} k q^2$$ + +**Action Integral** + +To find the path $q(t)$ that the system takes between times $t_1$ and $t_2$, we evaluate the action integral: +$$S[q(t)] = \int_{t_1}^{t_2} \left( \frac{1}{2} m \dot{q}^2 - \frac{1}{2} k q^2 \right) \, dt$$ + +**Finding the Path (Euler-Lagrange Equation)** + +Applying the **Euler-Lagrange equation** to the Lagrangian leads to the equation of motion: +$$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} = 0$$ + +Solving this gives the differential equation for a simple harmonic oscillator: +$$m \ddot{q} + k q = 0$$ + +which has solutions in the form of sinusoidal functions, describing the oscillatory motion $q(t)$ over time. + +If you squint, you'll see we've derived **Hooke's Law**. Since $\ddot{q} = a$ and $F=ma$: + +$$ ma = F = - kq $$ + +where $k$ is the _spring constant_ AKA restoring force over the distance ($\frac{N}{m}$). + +::: # Minkowski Space +> _Roads? Where we're going, we don't need roads_ [^3] + +[^3]: Dr. Emmmet Brown - Back to the Future ```{python} #| label: fig-minkowski_light_cone From eccfb8ab73ddd4bd33470dfa7ddc02bd1114c719 Mon Sep 17 00:00:00 2001 From: John Vivian Date: Sun, 3 Nov 2024 01:03:57 -0700 Subject: [PATCH 08/15] Expand on hamiltonian --- posts/classical-mechanics/index.qmd | 137 ++++++++++++++++++++++++++++ 1 file changed, 137 insertions(+) diff --git a/posts/classical-mechanics/index.qmd b/posts/classical-mechanics/index.qmd index a79be24..05d7a16 100644 --- a/posts/classical-mechanics/index.qmd +++ b/posts/classical-mechanics/index.qmd @@ -702,14 +702,103 @@ $$m \ddot{q} + k q = 0$$ which has solutions in the form of sinusoidal functions, describing the oscillatory motion $q(t)$ over time. +::: {.callout-warning collapse="true"} +## Hey not so fast, where the hell is $q(t)$? +Ok, ok sorry. I tried to pull a fast one. Solving the Lagrangian only gave us this differential equation—it didn’t directly give us the solution for +$q(t)$, which is the explicit path or trajectory of the system over time. + +We can rewrite our solution as + +$$\ddot{q} + \frac{k}{m}q=0$$ + + +Let $\omega = \sqrt{\frac{k}{m}}$, where $\omega$ is the **angular frequency** of the oscillator. Now the equation becomes: +$$\ddot{q} + \omega^2 q = 0$$ + +This equation has a general solution of the form: +$$q(t) = A \cos(\omega t) + B \sin(\omega t)$$ +where $A$ and $B$ are constants determined by the initial conditions of the system (such as initial position and velocity). + +::: {.callout-important collapse="true"} +## But wait, why is that the general solution? +Oh my god fuck off already +::: + +### Interpretation +This solution, $q(t) = A \cos(\omega t) + B \sin(\omega t)$, is the explicit **time-dependent path** or **trajectory** that the oscillator follows. It describes the oscillatory motion in terms of time, with: + +- $\omega$: The frequency of oscillation, showing how quickly the system oscillates. +- $A$ and $B$: Constants that depend on the starting conditions (e.g., where and how fast the mass was initially moving). +::: + If you squint, you'll see we've derived **Hooke's Law**. Since $\ddot{q} = a$ and $F=ma$: $$ ma = F = - kq $$ where $k$ is the _spring constant_ AKA restoring force over the distance ($\frac{N}{m}$). + ::: +## The Hamiltonian Reform(ul)ation +> Puttin' big E first + +We spent a a bit of time on Lagrangian mechanics because it also requires introduces several other interesting concepts like _the principle of least action_. The good news is we've done a lot of the heavy lifting and there's nothing too crazy about now thinking about Hamiltonian mechanics, where we shift from _paths_ to _energies_. + +While the Lagrangian $L = T - V$ is built around action, the Hamiltonian is a powerful reformulation that shifts the focus to energy. The Hamiltonian $H$ typically represents the total energy of a system $H = T + V$ (kinetic + potential energy) and is built to highlight conservation laws. + +Unlike the Lagrangian, the Hamiltonian only depends on the position $q$ and momentum $p$, but not directly on $t$ itself. This approach introduces **phase space** (a space defined by positions and momenta), allowing viusalization of a system’s entire state at any moment. Shifting from position/velocity to position/momentum reveals insights about stability, resonance, and even chaotic behavior in systems. + +Reviewing @fig-corrected_mechanics_comparison now may be elucidating. First, the phase space for the simple harmonic oscillator shows energy is conserved (closed ellipse). Second, thinking back to our ball-on-a-hill example and starting from the left and rolling down hill to the right, would hit maximum momentum at the bottom of the hill ($x=0$) and as it oscillated forever back-and-forth would follow a clockwise path around the ellipse (rolling back to where we started in this coordinate system means negative velocity therefore negative momentum). + +::: {.callout-note collapse="true"} +## Simple Harmonic Hamiltonian +> Here we can shortcut since we already did the Lagrangian solution. + +$$L = T - V = \frac{1}{2} m \dot{q}^2 - \frac{1}{2} k q^2$$ + +**Define the Hamiltonian $H = T + V$ in Terms of Position and Momentum** + +To switch to the Hamiltonian formalism, we need to express the system in terms of **position $q$** and **momentum $p$**, where momentum $p$ is defined as: +$$p = \frac{\partial L}{\partial \dot{q}} = m \dot{q}$$ + +Now, let’s express the **Hamiltonian** $H$, which for many systems is the **total energy** $T + V$: +$$H = T + V = \frac{p^2}{2m} + \frac{1}{2} k q^2$$ + +So, the Hamiltonian for a simple harmonic oscillator is: +$$H(q, p) = \frac{p^2}{2m} + \frac{1}{2} k q^2$$ + +**Interpret the Hamiltonian** + +The Hamiltonian here represents the **total energy** of the oscillator, with: + +- $\frac{p^2}{2m}$: The **kinetic energy** in terms of momentum $p$ +- $\frac{1}{2} k q^2$: The **potential energy** stored in the spring + +**Derive the Equations of Motion Using Hamilton’s Equations** + +In Hamiltonian mechanics, we use **Hamilton’s equations** to describe the time evolution of the system: + +$$\dot{q} = \frac{\partial H}{\partial p} \quad \text{and} \quad \dot{p} = -\frac{\partial H}{\partial q}$$ + +Applying these to our Hamiltonian: +1. **Find $\dot{q}$**: + $$\dot{q} = \frac{\partial H}{\partial p} = \frac{p}{m}$$ + This confirms our definition of momentum, $p = m \dot{q}$. + +2. **Find $\dot{p}$**: + $$\dot{p} = -\frac{\partial H}{\partial q} = -k q$$ + This gives us the familiar equation of motion for a simple harmonic oscillator: + $$m \ddot{q} + k q = 0$$ + +**Visualizing in Phase Space** + +The Hamiltonian approach naturally introduces **phase space**, where each point represents a state of the system with coordinates $(q, p)$. For a simple harmonic oscillator: +- The system’s trajectory in phase space forms an **ellipse**. +- This elliptical trajectory represents the **conservation of total energy**: as the oscillator moves, $q$ and $p$ oscillate back and forth, but the total energy (Hamiltonian) remains constant. + +This example highlights the power of the Hamiltonian formalism in tracking **conserved quantities** and understanding **system behavior** in phase space. +::: # Minkowski Space > _Roads? Where we're going, we don't need roads_ [^3] @@ -755,6 +844,54 @@ plt.show() ``` +```{python} +#| label: fig-pythag +#| fig-cap: Real Time in Minkowski space + +fig, ax = plt.subplots(figsize=(8, 8)) + +# Define the light cone (45-degree lines) for reference +x = np.linspace(-10, 10, 400) +t = np.abs(x) # Light moves at 45 degrees, so t = |x| + +# Plot the light cone as dashed lines +ax.plot(x, t, 'k--', alpha=0.5, label="Light Path (c)") +ax.plot(x, -t, 'k--', alpha=0.5) + +# Coordinates for an example object moving slower than light +x_object = 6 +t_object = 8 +s_object = np.sqrt((t_object)**2 - (x_object)**2) # Proper time (tau), the hypotenuse + +# Draw the object’s path +ax.plot([0, x_object], [0, t_object], color='blue', label="Object Path") +ax.plot([0, x_object], [0, 0], 'b:', linewidth=1) # Projected x-component +ax.plot([x_object, x_object], [0, t_object], 'b:', linewidth=1) # Projected t-component + +# Draw the hypotenuse representing real time (proper time) +ax.annotate(r"$\tau$ (Proper Time)", xy=(x_object/2, t_object/2), xytext=(x_object/2 + 1, t_object/2 + 1), + arrowprops=dict(facecolor='blue', arrowstyle="->"), fontsize=12, color='blue') + +# Label and annotate +ax.text(x_object + 0.5, t_object, r"$(x, t)$", fontsize=12, color="blue") +ax.text(x_object/2 + 1, -0.5, "Space (x)", ha="center", fontsize=12) +ax.text(0.5, t_object/2 + 1, "Time (t)", ha="center", rotation=90, fontsize=12) +ax.text(x_object - 3, t_object + 1, r"$s = \sqrt{c^2 t^2 - x^2}$", fontsize=12, color="blue") + +# Set axis limits and labels +ax.set_xlim(-10, 10) +ax.set_ylim(-10, 10) +ax.set_xlabel("Space (x)", fontsize=12) +ax.set_ylabel("Time (t)", fontsize=12) + +# Legend and display adjustments +ax.legend(loc="upper left") +ax.axhline(0, color='black',linewidth=0.5) +ax.axvline(0, color='black',linewidth=0.5) + +plt.show() +``` + ```{python} #| label: corrected_worldlines_lightcone #| fig-cap: "Minkowski Space: Worldlines for Stationary and Moving Observers" From ac1863fed0d19bd00d99c061703f082eb3d0c35a Mon Sep 17 00:00:00 2001 From: jvivian Date: Mon, 4 Nov 2024 22:33:16 -0800 Subject: [PATCH 09/15] Expand calculus sections, edit gravity section, add notes for final major section --- posts/classical-mechanics/index.qmd | 157 +++++++++++++++++++++++++--- 1 file changed, 142 insertions(+), 15 deletions(-) diff --git a/posts/classical-mechanics/index.qmd b/posts/classical-mechanics/index.qmd index 1aa30d1..e0b239e 100644 --- a/posts/classical-mechanics/index.qmd +++ b/posts/classical-mechanics/index.qmd @@ -25,7 +25,12 @@ plt.style.use('ggplot') > _I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me_ [^1] -This overview is roughly structured by the concepts I found most profound while reading and researching topics in Sean Carroll's book _The Biggest Ideas in The Universe: Space, Time, and Motion_. +[^1]: Isaac Newton + +::: {.callout-note} +## Motivation +This overview is roughly structured by the concepts I found most profound while reading and researching topics in Sean Carroll's book _The Biggest Ideas in The Universe: Space, Time, and Motion_. +::: - While velocity only has meaning in other reference frames, acceleration is _absolute_, and can be identified intriniscally without one (in most cases). - The squared term in the inverse square law is derived from the spacial geometry we live in (three dimensions). @@ -34,19 +39,22 @@ This overview is roughly structured by the concepts I found most profound while - Conservation laws arise directly from symmetries in the laws of physics. I.e. The invariance of a system's laws relate directly to a given conservation law: time invariance $\rightarrow$ energy conservation, spatial invariance $\rightarrow$ momentum conservation, and rotational symmetry $\rightarrow$ angular momentum conservation. - The speed of light, $c$, provides a conversion factor between time and space. Like energy $\Leftrightarrow$ mass and electricity $\Leftrightarrow$ magnetism, they are aspects of the same phenomena. - Lorentz transformations, which describe effects like length contraction and time dilation, can be derived from the invariant spacetime interval---analagous to the Pythagorean theorem---in the geometry of Minkowski space. -- Schwartzchild's solution to General Relativity contained a mathematical singularity -- originally thought to be a _coordinate singularity_ -- turned out to be a real _curvature singularity_ in spacetime: a blackhole. +- Schwartzchild's solution to General Relativity contained a curious mathematical singularity which -- originally thought to be a _coordinate singularity_ -- turned out to be a real _curvature singularity_ in spacetime: the infamous blackhole. + +::: {.callout-warning} +## Disclaimer +I’m neither a mathematician or physicist but do my best to clarify the meaning of each variable and operator in the equations introduced. +::: -**Disclaimer**: I’m not a mathematician or physicist, so aim to clarify the meaning of each variable and operator in the equations introduced. +# Absolute Power, Absolute Acceleration -# Acceleration is Absolute +Imagine yourself floating in deep space, with only distant stars as a backdrop. Would you be able to tell if you were moving? Surprisingly, no. As Einstein demonstrated, **velocity only has meaning relative to another reference frame**. If two people drift past each other, neither can determine who is “stationary” or “moving” in any absolute sense—both could be moving, or one could be stationary. In empty space, without other points of reference, velocity becomes entirely relative. -Imagine yourself in deep space, with only distant stars as a backdrop. Would it be possible to tell if you were moving? Nope. As Einstein showed, velocity only has meaning relative to another reference frame. If two people drifted past each other, neither could determine who was stationary or moving, or if they were both moving at once. +Oddly, _acceleration is different_. If you were suddenly accelerated in your spacesuit, you’d feel it immediately—your body would press against the suit, jerking you forward. Acceleration creates a force that acts on you directly, causing a _force gradient_ you can feel. Unlike velocity, which requires no force to sustain, acceleration always involves a force acting on a mass. -Imagine yourself in space with no nearby objects, only distant stars; would it be possible to tell you are moving? Nope. As Einstein later proved, velocity only has context relative to another reference frame. Two people moving in in the same environment past each other would not be able to conclude whether one person was stationary and the other was moving or if both people are moving. +In fact, you can create your own “accelerometer” with something as simple as a spring and weight. Hang the weight from a spring, and if you accelerate left or right, the weight will stretch the spring in the opposite direction. This setup works because acceleration produces a noticeable force, while constant velocity does not. -Oddly, _acceleration_ is not the same. If you were suddenly accelerated while in the space suit you would feel it; you would jerk forward, feel the suit push against your chest, etc. The reason for this is acceleration produces a _force gradient_, as opposed to velocity which does not require a force to keep you moving (an object in motion, yada yada...). -You can build your own accelerator-detector just by hanging a weight from a spring. If you begin accelerating left or right, the spring would stretch in the opposite direction. The weight needs to hang to work, because if you turn it sideways the weight will fall towards the ground, even if remain perfectly still. You _are_ accelerating, and have been for quite some time, directly towards the center of the planet. We experience this acceleration as _weight_, which is canceled out by the normal force of the planet pushing back up against us. ```{python} @@ -100,7 +108,13 @@ ax.legend(loc='upper right') plt.show() ``` -Physics equations are extraordinary because they describe _universal relationships_ in the natural world. For instance, the _momentum_ of an object is simply the product of its mass and velocity: $p = mv$. This relationship applies universally to every object with mass and velocity. And I think that's pretty neat [^2]. Notably, the equation shows that momentum is directly proportional to velocity for a constant mass—doubling an object’s velocity doubles its momentum. +Back on Earth, even when we’re standing still, we feel the effect of acceleration toward the planet’s center, which we experience as _weight_. This downward pull is balanced by the normal force of the ground pushing back up against us, allowing us to experience acceleration even when we appear stationary. + +--- + +Physics equations are extraordinary because they describe _universal relationships_ in the natural world. For instance, the **momentum** of an object is simply the product of its mass and velocity: +$$p = mv$$ +This relationship applies universally to every object with mass and velocity. And I think that's pretty neat [^2]. Notably, the equation shows that momentum is directly proportional to velocity for a constant mass—doubling an object’s velocity doubles its momentum. [^2]: Neature Walk: https://www.youtube.com/watch?v=Hm3JodBR-vs @@ -125,12 +139,64 @@ An interesting consequence to these equations is given position or velocity as a ::: {.callout-tip collapse="true"} ## What's the point of calculus -Calculus solves two limit-based problems: +Calculus addresses two fundamental problems involving limits: **rates of change** and **accumulation**. + +**Derivatives**: The derivative represents the rate of change of a quantity. It answers questions like, “How fast is something changing at an exact moment?” Calculus solves this by finding $\frac{dx}{dt}$ as both $dx$ and $dt$ approach zero—a challenge because dividing by zero is undefined. By using limits, calculus lets us precisely calculate this “instantaneous” rate without ever actually dividing by zero. + +**Integrals**: The integral represents the accumulation of quantities over an interval, like the area under a curve. To do this, calculus sums up infinitely many tiny pieces (Riemann squares) as their width approaches zero. This limit-based approach lets us find the exact area, even under curves that are difficult or impossible to sum otherwise. + + +```{python} +#| label: fig-calculus +#| fig-cap: Given some continuous function, the derivative is _instantaneous rate of change_ at a given point. +# Set up a 1x2 subplot for Derivative and Integral plots side by side +fig, axes = plt.subplots(2, 1, figsize=(6, 5)) +x = np.linspace(0, 10, 100) + +# Define a quadratic function for position over time, e.g., f(x) = 0.5*x^2 (constant acceleration) +position = 0.5 * x**2 + +# Derivative Plot (Position by time with tangent line) +axes[0].plot(x, position, label=r"$f(x) = \frac{1}{2}x^2$", color="blue") +tangent_x = 5 # Point at which to draw the tangent +tangent_slope = tangent_x # Derivative of 0.5*x^2 is x, so slope at x = 5 is 5 +tangent_line = tangent_slope * (x - tangent_x) + 0.5 * tangent_x**2 +axes[0].plot(x, tangent_line, linestyle="--", color="orange", label=r"$\frac{dx}{dt}$ at $x=5$") +axes[0].scatter(tangent_x, 0.5 * tangent_x**2, color="red") # Point of tangency +axes[0].set_title("Derivative (Instantaneous Rate of Change)") +axes[0].set_xlabel("Time (t)") +axes[0].set_ylabel("Position (f(x))") +axes[0].legend() +axes[0].grid(True, linestyle=':', alpha=0.7) + +# Integral Plot (Riemann squares for area under the curve) +f_x = 0.5 * x**2 +num_squares = 10 # Number of Riemann squares +x_riemann = np.linspace(0, 10, num_squares + 1) +y_riemann = 0.5 * x_riemann**2 +widths = x_riemann[1] - x_riemann[0] + +# Plot the position curve for reference +axes[1].plot(x, f_x, color="blue", label=r"$f(x) = \frac{1}{2}x^2$") +# Plot Riemann squares +for i in range(num_squares): + axes[1].bar(x_riemann[i], y_riemann[i], width=widths, align="edge", color="orange", edgecolor="black", alpha=0.3) + +axes[1].set_title("Integral (Riemann Sum Approximation)") +axes[1].set_xlabel("x") +axes[1].set_ylabel("f(x)") +axes[1].legend() +axes[1].grid(True, linestyle=':', alpha=0.7) + +# Layout adjustment +plt.tight_layout() +plt.show() +``` + -1. The goal of the derivative is to find $\frac{dx}{dt}$ as they both approach zero, but you can't divide by zero. -2. The integral attempts to find the sum of the area under the curve as the number of riemann squares approaches infinity. +For example, if we expand a small change in a function, such as $(x + h)^2 - x^2$, and take the limit as $h \rightarrow 0$, we end up with $2x$—the derivative of $x^2$. What's really magical is the inverse of $x^2$ is the integral. This makes sense if you consider the derivative gives the rate of change (e.g. velocity from a function of position) and integration adds the accumulated change to get the original value (e.g. total distance from a function of velocity). -You can expand a small change in a simple function (such as $x^2$, i.e. $(x+h)^2 - x^2$) and work out as the limit $h \rightarrow 0$, you get $2x$. +$$ \frac{dx}{dt} f(x) = \int f(v) \space dt $$ ::: :::{.callout-note collapse="true"} @@ -164,6 +230,20 @@ $$v^2 = v_0^2 + 2a(x-x0)$$ ::: ## The Universal Law of Gravitation +> Quote here + +Newton’s Law of Gravitation describes the gravitational force between two masses, expressing how every mass exerts an attractive force on every other mass in the universe: + +$$ +F = G\frac{m_1 m_2}{r^2} +$$ + +where: + +- $F$ is the gravitational force between the two masses. +- $G$ is the gravitational constant, a fundamental constant that calibrates the strength of gravity. +- $m_1$ and $m_2$ are the masses of the two objects. +- $r$ is the distance between the centers of the two masses. ```{python} #| label: fig-gravitational_force_visualization @@ -215,6 +295,13 @@ ax.legend(loc='upper right') plt.show() ``` +### Why the Inverse-Square Law? Why $r^2$? +The inverse-square relationship, $\frac{1}{r^2}$, reflects a fundamental property of how forces and fields spread out in three-dimensional space. Newton’s insight into this came from observing that gravitational and other forces, like light intensity, diminish in strength with the square of the distance. + +In three-dimensional space, when a force radiates outward from a point source (such as a mass), it spreads uniformly over the surface of an expanding sphere centered around that source. The surface area of a sphere grows proportionally to $r^2$, where $r$ is the radius of the sphere. Consequently, as the distance $r$ increases, the same force is distributed over a larger area, causing the intensity at any given point on that surface to diminish in proportion to $\frac{1}{r^2}$. + +Thus, the inverse-square law isn’t arbitrary but rather a natural consequence of geometry in three-dimensional space. If we lived in a universe with different spatial dimensions, the exponent would change accordingly. For example, in a two-dimensional plane, the force would weaken with $\frac{1}{r}$, while in four dimensions, it would decay with $\frac{1}{r^3}$. + Newton's Universal Law of Gravitation describes the generated by two masses. This equation also ties together the inverse square law (decay of intensity of effects such as light, radiation, or gravity). $$F = G\frac{m_1m_2}{r^2}$$ @@ -479,7 +566,7 @@ plt.tight_layout() plt.show() ``` -### The Bigger Picture: Electromagnetic Waves +### The Unification of Electromagnetism Maxwell’s equations, taken together, explain how electric and magnetic fields interact and propagate. If you combine Faraday's Law and Ampère’s Law, you can derive the **wave equation** for both electric and magnetic fields, showing that disturbances in these fields propagate at the speed of light. Together, these two effects allow electromagnetic waves to propagate through space, with the electric and magnetic fields continuously regenerating each other. This is why light, radio waves, X-rays, and all other electromagnetic radiation are essentially **electromagnetic waves**: oscillating electric and magnetic fields that move through space, even in a vacuum. @@ -796,11 +883,27 @@ The Hamiltonian approach naturally introduces **phase space**, where each point This example highlights the power of the Hamiltonian formalism in tracking **conserved quantities** and understanding **system behavior** in phase space. ::: -# Minkowski Space +# Pythagoras in Minkowski Space > _Roads? Where we're going, we don't need roads_ [^3] [^3]: Dr. Emmmet Brown - Back to the Future +## Special Relativity and the Insanity of $c$ + +Goal: Introduce special relativity by focusing on why $c$, the speed of light, is fundamental and invariant. + +Key Points: The postulate that $c$ is the same for all observers leads to time dilation and length contraction. +Explain how $c$ creates a relationship between space and time, where distance can be "subtracted" from time, giving rise to the spacetime interval. + +Connection: This section sets up Minkowski space and real time, motivating the need for a geometric interpretation of spacetime. + +## Lost In (Minkowski) Space +Goal: Introduce the geometry of Minkowski space and the concept of the spacetime interval. + +Key Points: Define Minkowski space and the spacetime interval: $s^2 = c^2t^2 - x^2$. Explain that the interval $s$ remains invariant for all observers and represents real time (proper time), which serves as the "distance" in this geometry. + +Connection: Introduce the idea that spacetime can be curved by mass, which lays the groundwork for general relativity and the Schwarzschild solution. + ```{python} #| label: fig-minkowski_light_cone #| fig-cap: Light Cone in Minkowski Space. A stationary observer moves vertically along a 'World Line' @@ -939,7 +1042,31 @@ plt.show() ``` +## General Relativity: Gravity as Curvature +> May the Schwarz(schild Solution) be with you [^4] + +[^4]: Space Balls - 1987 + +Goal: Introduce general relativity’s core idea that mass and energy curve spacetime, with a specific focus on the Schwarzschild solution. + +Key Points: Briefly explain how Einstein’s field equations describe how mass curves spacetime. Describe the Schwarzschild solution as a specific, spherically symmetric solution that describes the geometry outside a massive, non-rotating object. Explain the Schwarzschild radius and its role in defining the event horizon. + +Connection: Use this to introduce black holes and explain why crossing the event horizon changes the nature of time and space. + +## Black Holes: The Inevitable Exit from Our Universe +Goal: Define black holes, focusing on the event horizon and the singularity. + +Key Points: Distinguish between the event horizon (boundary from which light cannot escape) and the singularity (where spacetime curvature becomes infinite). Describe how the event horizon represents a “point of no return”—once crossed, all paths lead to the singularity. + +Connection: Lead into a discussion of Kruskal-Szekeres coordinates as a way to better understand spacetime inside the horizon. + +## Kruskal-Szekeres Coordinates: Inside the Event Horizon + +Goal: Use Kruskal coordinates to explain why the singularity lies in the future of any object crossing the event horizon. + +Key Points:Explain how Kruskal coordinates extend the Schwarzschild solution, allowing us to visualize the paths of objects inside the event horizon. Describe how, inside the event horizon, the time and space axes effectively “flip”—$t$ becomes spatial, and $x$ becomes temporal. Discuss the inevitability of the singularity: once inside, all directions in time lead to the singularity. +Connection: This section ties together how both special and general relativity redefine our notions of space, time, and causality. # Glossary From 35cce261b3eb7b88af6591f56fb17c074f199827 Mon Sep 17 00:00:00 2001 From: jvivian Date: Tue, 5 Nov 2024 13:13:55 -0800 Subject: [PATCH 10/15] Update relativity and minkowski space-time --- posts/classical-mechanics/index.qmd | 75 ++++++++++++++++++++++++----- 1 file changed, 62 insertions(+), 13 deletions(-) diff --git a/posts/classical-mechanics/index.qmd b/posts/classical-mechanics/index.qmd index e0b239e..62805eb 100644 --- a/posts/classical-mechanics/index.qmd +++ b/posts/classical-mechanics/index.qmd @@ -1,8 +1,9 @@ --- -title: "A Journey Through (Space)Time" -subtitle: "Summary of Classical Mechanics" +title: "Pythagorean Time Dilation" +subtitle: "A Summary of Classical Mechanics" author: John Vivian date: 2024-10-15 +categories: [Math, Physics] format: html: # css: styles.css @@ -23,15 +24,17 @@ import seaborn as sns plt.style.use('ggplot') ``` -> _I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me_ [^1] +# A Journey Through (Space)Time -[^1]: Isaac Newton +> _I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me_ [^1] ::: {.callout-note} ## Motivation This overview is roughly structured by the concepts I found most profound while reading and researching topics in Sean Carroll's book _The Biggest Ideas in The Universe: Space, Time, and Motion_. ::: +[^1]: Isaac Newton + - While velocity only has meaning in other reference frames, acceleration is _absolute_, and can be identified intriniscally without one (in most cases). - The squared term in the inverse square law is derived from the spacial geometry we live in (three dimensions). - Magnetism only exists as a relativistic effect when an object moves relative to an electric field, which helps explain why we have never observed a magnetic monopole. @@ -39,7 +42,7 @@ This overview is roughly structured by the concepts I found most profound while - Conservation laws arise directly from symmetries in the laws of physics. I.e. The invariance of a system's laws relate directly to a given conservation law: time invariance $\rightarrow$ energy conservation, spatial invariance $\rightarrow$ momentum conservation, and rotational symmetry $\rightarrow$ angular momentum conservation. - The speed of light, $c$, provides a conversion factor between time and space. Like energy $\Leftrightarrow$ mass and electricity $\Leftrightarrow$ magnetism, they are aspects of the same phenomena. - Lorentz transformations, which describe effects like length contraction and time dilation, can be derived from the invariant spacetime interval---analagous to the Pythagorean theorem---in the geometry of Minkowski space. -- Schwartzchild's solution to General Relativity contained a curious mathematical singularity which -- originally thought to be a _coordinate singularity_ -- turned out to be a real _curvature singularity_ in spacetime: the infamous blackhole. +- Schwartzchild's solution to General Relativity contained a curious mathematical singularity which---originally thought to be a _coordinate singularity_---turned out to be a real _curvature singularity_ in spacetime: the infamous **blackhole**. ::: {.callout-warning} ## Disclaimer @@ -890,19 +893,30 @@ This example highlights the power of the Hamiltonian formalism in tracking **con ## Special Relativity and the Insanity of $c$ -Goal: Introduce special relativity by focusing on why $c$, the speed of light, is fundamental and invariant. +At 16 years old Einstein had thoughts about what it would be like to chase down a beam of light coming out of a flashlight. Classically, if he moved fast enough he could coast alongside that wave of light, like when a train matches the speed of a passing bird. + +An unexpected consequence of the **Michelson-Morley Experiment** of 1887---while looking for a theoretical "luminiferous aether"---was that the inferometer detected light moving the same speed regardless of direction, suggesting it was the same speed regardless of how the Earth moved through space, therefore **independent of the observer's motion**. This led the groundwork for Einstein's theory. + +### The Speed of Light, $c$, and the Foundations of Special Relativity -Key Points: The postulate that $c$ is the same for all observers leads to time dilation and length contraction. -Explain how $c$ creates a relationship between space and time, where distance can be "subtracted" from time, giving rise to the spacetime interval. +Einstein realized that, to be consistent with Maxwell’s equations for electromagnetism, light’s speed must remain constant across all frames of reference. This insight forced a rethinking of space and time. The concepts of absolute time (a universal “clock”) and absolute space (a universal grid) had to be discarded. Instead, space and time are intertwined in such a way that the speed of light remains constant—a revolutionary shift that led to the notion of spacetime. -Connection: This section sets up Minkowski space and real time, motivating the need for a geometric interpretation of spacetime. +1. Invariance of $c$: The speed of light in a vacuum is the same for all observers, regardless of their motion. This means that whether you’re standing still, driving a car, or hurtling through space, $c$ is always the same. +2. Space and Time Adapt: For this invariance to hold, space and time themselves must adapt to keep the speed of light constant. If you move faster, time slows down and lengths contract to ensure you never catch up with light. +3. Relativity of Simultaneity: Events that appear simultaneous to one observer may not be simultaneous to another moving observer. This idea challenges our intuitive notion of a shared “now” and suggests that time flows differently depending on motion. Ab event in one observer's past can be another's future depending on how fast they are moving relative to one another. +4. More directly, we now know this also limits _informations_ ability to propagate faster than speed of light. Quantum entanglement, counter-intuitively, does _not_ violate this principle. -## Lost In (Minkowski) Space -Goal: Introduce the geometry of Minkowski space and the concept of the spacetime interval. +### Time is Just Another Coordinate -Key Points: Define Minkowski space and the spacetime interval: $s^2 = c^2t^2 - x^2$. Explain that the interval $s$ remains invariant for all observers and represents real time (proper time), which serves as the "distance" in this geometry. +In everyday life, we already treat time and space as coordinates, although we often don’t think of them this way. Imagine you’re meeting a friend at a café. You naturally specify a location (**space coordinates**) and a time (**time coordinate**). Without both, you might arrive at the café only to miss each other by an hour or by a block. -Connection: Introduce the idea that spacetime can be curved by mass, which lays the groundwork for general relativity and the Schwarzschild solution. +Einstein’s insight was that this relationship goes much deeper: **space and time are not separate entities but are interwoven**. Just as you can’t meet your friend without specifying both space and time, the universe itself weaves these dimensions together in a way that affects how time flows and distances change. + +We can now measure something called the **spacetime interval** that takes both space and time into account by plotting space against time and exploring how $c$ creates boundaries within this space and how spacetime intervals (measured as the hypotenuse on this graph) remain constant for all observers (and lead to the derivations of the **Lorentz Transformations**). + +## Lost In (Minkowski) Space: The Geometry of Spacetime + +In everyday life, we think of **space** and **time** as separate. We measure distances in meters or miles and keep track of time with clocks. However, special relativity showed us that space and time are deeply connected, forming a unified framework called **spacetime**. We can described this framework mathematically using a special kind of geometry called **Minkowski space**. ```{python} #| label: fig-minkowski_light_cone @@ -943,6 +957,12 @@ plt.show() ``` +### The Spacetime Interval + +In Minkowski space, the fundamental unit of "distance" isn’t a simple measurement of space or time alone—it’s a combination of both, called the **spacetime interval**. For two events separated by a distance $x$ and a time $t$, the spacetime interval $s$ is given by: + +$$s^2 = c^2 t^2 - x^2$$ + ```{python} #| label: fig-pythag #| fig-cap: Real Time in Minkowski space @@ -991,6 +1011,35 @@ ax.axvline(0, color='black',linewidth=0.5) plt.show() ``` + +This interval differs from the usual Euclidean distance formula. + +- $c$ is the speed of light, a constant that connects space and time. +- The interval $s$ combines spatial separation ($x$) and time separation ($t$) but with a **minus sign** in front of $x^2$, which makes the geometry of Minkowski space different from ordinary space. +- Note how we keep units consistent by multiplying $t$ by $c$. We're not in Kansas anymore [^5]. + +[^5]: The Wizard of Oz + +### Why This Interval Matters + +The spacetime interval $s$ has an important property: it’s **invariant** for all observers, regardless of their relative motion. This invariance means that while two observers might disagree on measurements of $x$ and $t$, they will always agree on the value of $s$. In other words, even though space and time are experienced differently depending on motion, **spacetime itself has an unchanging “distance”** between events. + +This interval $s$ is also known as **proper time** when it represents the "real time" experienced by an object moving along a specific path in spacetime. Just as distance measures separation in space, the spacetime interval $s$ measures separation in spacetime. + +### Visualizing Minkowski Space + +In Minkowski space, we can plot time and space as perpendicular axes, similar to a regular coordinate system, but with some key differences. Points (or “events”) in spacetime are located by both space and time coordinates. If we connect two events, the **spacetime interval** $s$ between them becomes the “distance” in this spacetime geometry. + +This interval sets limits on causality: events separated by a spacetime interval greater than zero (i.e., spacelike intervals) cannot influence each other, as no signal can travel faster than the speed of light. Light itself defines the **boundary of possible influence**, forming a "light cone" structure in this space. + +### Toward General Relativity: Curved Spacetime + +Minkowski space describes a flat spacetime, where the geometry remains consistent everywhere. But what happens when we introduce **mass**? Einstein’s theory of general relativity builds on this framework by showing that **mass curves spacetime**. Massive objects like planets and stars create distortions, bending the paths of objects and even light. This curved geometry replaces Minkowski space with a more complex spacetime that adapts to the distribution of mass and energy. + +This concept of curved spacetime is described mathematically by the **Schwarzschild solution**, a specific solution to Einstein’s field equations. It describes the geometry around a massive, non-rotating object, leading us to concepts like **black holes** and **event horizons**. + +In summary, Minkowski space provides the foundation for understanding spacetime as a unified geometry where space and time are linked through the spacetime interval. This flat spacetime model prepares us for the next step: seeing how spacetime itself bends in the presence of mass, laying the groundwork for the profound ideas of general relativity. + ```{python} #| label: corrected_worldlines_lightcone #| fig-cap: "Minkowski Space: Worldlines for Stationary and Moving Observers" From b0d73cf7f157e20dfd9a26c6264cb16b8632c42f Mon Sep 17 00:00:00 2001 From: jvivian Date: Tue, 5 Nov 2024 19:53:35 -0800 Subject: [PATCH 11/15] Add section on time dilation / length contraction derivations --- posts/classical-mechanics/index.qmd | 234 ++++++++++++++++++++++------ 1 file changed, 183 insertions(+), 51 deletions(-) diff --git a/posts/classical-mechanics/index.qmd b/posts/classical-mechanics/index.qmd index 62805eb..0b42b08 100644 --- a/posts/classical-mechanics/index.qmd +++ b/posts/classical-mechanics/index.qmd @@ -902,8 +902,8 @@ An unexpected consequence of the **Michelson-Morley Experiment** of 1887---while Einstein realized that, to be consistent with Maxwell’s equations for electromagnetism, light’s speed must remain constant across all frames of reference. This insight forced a rethinking of space and time. The concepts of absolute time (a universal “clock”) and absolute space (a universal grid) had to be discarded. Instead, space and time are intertwined in such a way that the speed of light remains constant—a revolutionary shift that led to the notion of spacetime. 1. Invariance of $c$: The speed of light in a vacuum is the same for all observers, regardless of their motion. This means that whether you’re standing still, driving a car, or hurtling through space, $c$ is always the same. -2. Space and Time Adapt: For this invariance to hold, space and time themselves must adapt to keep the speed of light constant. If you move faster, time slows down and lengths contract to ensure you never catch up with light. -3. Relativity of Simultaneity: Events that appear simultaneous to one observer may not be simultaneous to another moving observer. This idea challenges our intuitive notion of a shared “now” and suggests that time flows differently depending on motion. Ab event in one observer's past can be another's future depending on how fast they are moving relative to one another. +2. Space and Time Adapt: For this invariance to hold, space and time themselves must adapt to keep the speed of light constant. If you move faster, your time slows down and lengths contract relative to slower-moving observers. +3. Relativity of Simultaneity: Events that appear simultaneous to one observer may not be simultaneous to another moving observer. This idea challenges the intuitive notion of a shared “now” and suggests that time flows differently depending on motion. An event in one observer's past can be in another's future depending on how fast they are moving relative to one another. 4. More directly, we now know this also limits _informations_ ability to propagate faster than speed of light. Quantum entanglement, counter-intuitively, does _not_ violate this principle. ### Time is Just Another Coordinate @@ -916,7 +916,7 @@ We can now measure something called the **spacetime interval** that takes both s ## Lost In (Minkowski) Space: The Geometry of Spacetime -In everyday life, we think of **space** and **time** as separate. We measure distances in meters or miles and keep track of time with clocks. However, special relativity showed us that space and time are deeply connected, forming a unified framework called **spacetime**. We can described this framework mathematically using a special kind of geometry called **Minkowski space**. +We can described the unified spacetime framework mathematically using a special kind of geometry called **Minkowski space**. In this space, all objects project a **causal light cone** forward along the time axis that restricts what other events in spacetime they can interact with. Photons, which are massless, propagate along the diagonals $y = \pm \space x$. ```{python} #| label: fig-minkowski_light_cone @@ -959,7 +959,7 @@ plt.show() ### The Spacetime Interval -In Minkowski space, the fundamental unit of "distance" isn’t a simple measurement of space or time alone—it’s a combination of both, called the **spacetime interval**. For two events separated by a distance $x$ and a time $t$, the spacetime interval $s$ is given by: +In Minkowski space, the fundamental unit of "distance" isn’t a simple measurement of space or time alone—it’s a combination of both, called the **spacetime interval**. For two events separated by a distance $x$ and a time $t$, the spacetime interval $s$ is given by a modified form of the Pythagorean Theorem: $$s^2 = c^2 t^2 - x^2$$ @@ -1011,86 +1011,218 @@ ax.axvline(0, color='black',linewidth=0.5) plt.show() ``` - This interval differs from the usual Euclidean distance formula. - $c$ is the speed of light, a constant that connects space and time. - The interval $s$ combines spatial separation ($x$) and time separation ($t$) but with a **minus sign** in front of $x^2$, which makes the geometry of Minkowski space different from ordinary space. - Note how we keep units consistent by multiplying $t$ by $c$. We're not in Kansas anymore [^5]. -[^5]: The Wizard of Oz +[^5]: Dorothy - The Wizard of Oz ### Why This Interval Matters The spacetime interval $s$ has an important property: it’s **invariant** for all observers, regardless of their relative motion. This invariance means that while two observers might disagree on measurements of $x$ and $t$, they will always agree on the value of $s$. In other words, even though space and time are experienced differently depending on motion, **spacetime itself has an unchanging “distance”** between events. -This interval $s$ is also known as **proper time** when it represents the "real time" experienced by an object moving along a specific path in spacetime. Just as distance measures separation in space, the spacetime interval $s$ measures separation in spacetime. +This interval $s$ is also known as **proper time** when it represents the "real time" experienced by an object moving along a specific path in spacetime. Just as distance measures separation in space, the spacetime interval $s$ measures separation in spacetime. -### Visualizing Minkowski Space +One unintuitive concept about Minkowski space is: due to the negative sign in how the spacetime interval is calculated, a longer line does not correspond to _more time experienced_. In fact, just the opposite is true. Relative to a stationary reference view, the closer on object moves to the speed of light the less time it will experience (**Time Dilation**) and the smaller the object will appear (**Length Contraction**). -In Minkowski space, we can plot time and space as perpendicular axes, similar to a regular coordinate system, but with some key differences. Points (or “events”) in spacetime are located by both space and time coordinates. If we connect two events, the **spacetime interval** $s$ between them becomes the “distance” in this spacetime geometry. +::: {.callout-note} +## Derivation of Lorentz Transformations from Spacetime Interval +To derive the time dilation and length contraction formulas from the spacetime interval, we can start with the invariant interval in special relativity. The spacetime interval $s^2$ between two events in Minkowski space is given by: -This interval sets limits on causality: events separated by a spacetime interval greater than zero (i.e., spacelike intervals) cannot influence each other, as no signal can travel faster than the speed of light. Light itself defines the **boundary of possible influence**, forming a "light cone" structure in this space. +$$ +s^2 = c^2 \Delta t^2 - \Delta x^2 +$$ -### Toward General Relativity: Curved Spacetime +where: -Minkowski space describes a flat spacetime, where the geometry remains consistent everywhere. But what happens when we introduce **mass**? Einstein’s theory of general relativity builds on this framework by showing that **mass curves spacetime**. Massive objects like planets and stars create distortions, bending the paths of objects and even light. This curved geometry replaces Minkowski space with a more complex spacetime that adapts to the distribution of mass and energy. +- $c$ is the speed of light, +- $\Delta t$ is the time difference between the two events in one frame, +- $\Delta x$ is the spatial separation between the events in the same frame. -This concept of curved spacetime is described mathematically by the **Schwarzschild solution**, a specific solution to Einstein’s field equations. It describes the geometry around a massive, non-rotating object, leading us to concepts like **black holes** and **event horizons**. +This interval $s^2$ is invariant (it remains the same) in all inertial reference frames and is the basis for both derivations. -In summary, Minkowski space provides the foundation for understanding spacetime as a unified geometry where space and time are linked through the spacetime interval. This flat spacetime model prepares us for the next step: seeing how spacetime itself bends in the presence of mass, laying the groundwork for the profound ideas of general relativity. +--- -```{python} -#| label: corrected_worldlines_lightcone -#| fig-cap: "Minkowski Space: Worldlines for Stationary and Moving Observers" -import numpy as np -import matplotlib.pyplot as plt -import seaborn as sns +### 1. Deriving Time Dilation +Consider two events that occur at the same location in a moving frame $S'$ (e.g., a clock moving with speed $v$ relative to an observer in frame $S$). -# Create figure and axis -fig, ax = plt.subplots(figsize=(8, 8)) +#### Setup: +- In the moving frame $S'$, the spatial separation $\Delta x' = 0$ because both events happen at the same place. +- Let $\Delta t'$ be the time interval between the two events in $S'$ (the proper time). +- In the stationary frame $S$, let the events be separated by a time interval $\Delta t$ and a spatial distance $\Delta x$. -# Generate time values -t = np.linspace(-5, 5, 100) +#### Interval in Frame $S'$: +Since $\Delta x' = 0$, +$$ +s^2 = c^2 \Delta t'^2 +$$ -# Plot light cone (x = ±t) -ax.plot(t, t, '--', color='orange', label='Light Cone: $x = ct$') -ax.plot(t, -t, '--', color='orange') +#### Interval in Frame $S$: +In the stationary frame $S$, the events are separated in both space and time: +$$ +s^2 = c^2 \Delta t^2 - \Delta x^2 +$$ -# Plot stationary observer's worldline (x = 0) -ax.plot([0, 0], [-5, 5], color='blue', lw=2, label='Stationary Observer (v = 0)') +Since $s^2$ is invariant, we equate the two expressions: +$$ +c^2 \Delta t'^2 = c^2 \Delta t^2 - \Delta x^2 +$$ -# Plot worldline for v = 0.3c -v1 = 0.3 # Velocity = 0.3c -x1 = v1 * t # Worldline for v = 0.3c -ax.plot(x1, t, '--', color='blue', lw=2, label='Worldline (v = 0.3c)', alpha=0.75) +Now, using the relationship $\Delta x = v \Delta t$ (where $v$ is the relative velocity of the two frames), +$$ +c^2 \Delta t'^2 = c^2 \Delta t^2 - (v \Delta t)^2 +$$ -# Plot worldline for v = 0.9c -v3 = 0.6 # Velocity = 0.9c -x3 = v3 * t # Worldline for v = 0.9c -ax.plot(x3, t, '--', color='blue', lw=2, label='Worldline (v = 0.6c)', alpha=0.5) +Factoring out $\Delta t^2$, +$$ +c^2 \Delta t'^2 = \Delta t^2 (c^2 - v^2) +$$ -# Plot worldline for v = 0.9c -v2 = 0.9 # Velocity = 0.9c -x2 = v2 * t # Worldline for v = 0.9c -ax.plot(x2, t, '--', color='blue', lw=2, label='Worldline (v = 0.9c)', alpha=0.25) +Dividing both sides by $c^2$, +$$ +\Delta t'^2 = \Delta t^2 \left( 1 - \frac{v^2}{c^2} \right) +$$ -# Set plot limits, labels, and title -ax.set_xlim(-5, 5) -ax.set_ylim(-5, 5) -ax.set_xlabel('Space ($x$)') -ax.set_ylabel('Time ($t$)') -ax.set_title('Minkowski Space: Worldlines for Different Velocities') +Taking the square root, +$$ +\Delta t' = \Delta t \sqrt{1 - \frac{v^2}{c^2}} +$$ -# Add legend -ax.legend(loc='upper left') +Since $\Delta t'$ is the proper time (the time in the frame where the events happen at the same location), we rearrange this to express $\Delta t$ in terms of $\Delta t'$: +$$ +\Delta t = \frac{\Delta t'}{\sqrt{1 - \frac{v^2}{c^2}}} +$$ -# Show plot -plt.show() +This is the **time dilation formula**. It shows that the time interval $\Delta t$ in the stationary frame $S$ (observed time) is longer than the time interval $\Delta t'$ in the moving frame $S'$ (proper time). In other words, a moving clock runs slower when observed from a stationary frame. + +--- + +### 2. Deriving Length Contraction + +Consider two points at rest in frame $S$ that define the endpoints of a rod, with a length $L$ measured in the frame $S$. + +#### Setup: +- In the stationary frame $S$, the endpoints of the rod are separated by $\Delta x = L$. +- In the moving frame $S'$, the endpoints are separated by $\Delta x' = L'$, which we want to find. +- The events marking the locations of the endpoints are simultaneous in $S$ (so $\Delta t = 0$ in $S$). + +#### Interval in Frame $S$: +Since the events are simultaneous in $S$, +$$ +s^2 = -\Delta x^2 = -L^2 +$$ + +#### Interval in Frame $S'$: +In the moving frame $S'$, where the rod is moving with velocity $v$, the spatial separation is $\Delta x' = L'$, and there may be a nonzero time difference $\Delta t'$ between the events. + +Using the invariant interval, +$$ +s^2 = c^2 \Delta t'^2 - L'^2 +$$ + +Equating the intervals, +$$ +-L^2 = c^2 \Delta t'^2 - L'^2 +$$ + +where $L$ is the length of the rod in the stationary frame $S$, and $L'$ is the length measured in the moving frame $S'$. To proceed, let’s set up the interval conditions a bit more carefully. + +#### Clarify the Setup +In the stationary frame $S$: + +- The events marking the endpoints of the rod are simultaneous ($\Delta t = 0$). +- The length $L$ is the proper length, as it’s measured in the frame where the rod is at rest. + +In the moving frame $S'$: + +- The endpoints are separated by $\Delta x' = L'$. +- Since the events are not simultaneous in $S'$, there may be a time difference $\Delta t'$ between observing each endpoint in $S'$. + +#### Interval Invariance Between Frames + +From the invariance of the spacetime interval, we have: +$$ +s^2 = -L^2 = c^2 \Delta t'^2 - L'^2 +$$ + +#### Finding $L'$ in Terms of $L$ +Since we want to measure the length of the rod in $S'$ (the moving frame), we must consider the relativity of simultaneity. For length contraction, we want to eliminate $\Delta t'$ because in $S'$ the length $L'$ is measured at a single moment (i.e., events should be simultaneous in $S'$ to define the length). + +To eliminate $\Delta t'$, use the fact that $\Delta t'$ is related to the motion of the rod. Given $\Delta x = L$ in $S$, and knowing that $S'$ moves at velocity $v$ relative to $S$, the time interval in $S'$ can be expressed as: +$$ +\Delta t' = \frac{v \Delta x}{c^2} = \frac{v L}{c^2} +$$ + +Substitute $\Delta t' = \frac{v L}{c^2}$ back into the interval equation: +$$ +-L^2 = c^2 \left( \frac{v L}{c^2} \right)^2 - L'^2 +$$ + +Simplify: +$$ +-L^2 = \frac{v^2 L^2}{c^2} - L'^2 +$$ + +Rearrange to solve for $L'^2$: +$$ +L'^2 = L^2 \left(1 - \frac{v^2}{c^2}\right) +$$ + +Taking the square root: +$$ +L' = L \sqrt{1 - \frac{v^2}{c^2}} +$$ + +### Final Length Contraction Formula +This is the **length contraction formula**: +$$ +L' = L \sqrt{1 - \frac{v^2}{c^2}} +$$ + +where: +- $L$ is the proper length (length of the rod in the frame where it is at rest), +- $L'$ is the contracted length as observed in the frame moving at speed $v$ relative to the rod. + +This shows an observer moving relative to the rod measures a shorter length than an observer at rest with respect to it. +::: + +```{python} +#| label: fig-length-contraction +#| fig-cap: Example of how long 1 meter rod would appear to an observer as they moved at different fractions of the speed of light +v_values = np.linspace(0, 0.99, 500) # velocities as fraction of c, up to 99% of c +L0 = 1 # initial (proper) length in meters + +# Length contraction formula +L_contracted = L0 * np.sqrt(1 - v_values**2) + +# Plot the contraction of length as velocity increases +fig, ax = plt.subplots(figsize=(8, 6)) +ax.plot(v_values, L_contracted, color='blue', label="Contracted Length") + +# Labels and plot styling +ax.set_xlabel("Velocity as a fraction of the speed of light (v/c)") +ax.set_ylabel("Observed Length (m)") +ax.set_title("Length Contraction of a 1 Meter Object vs. Speed") +ax.legend() +ax.grid(True) + +plt.show() ``` + +### Mass Curvature +> _...And I cannot lie.
You other brothers can't deny._ [^6] + +[^6]: Sir Mix-a-Lot - I Like Big Butts + +Minkowski space describes a flat spacetime, where the geometry remains consistent everywhere. But what happens when **mass** is introduced? Einstein’s theory of general relativity builds on his previous framework by realizing **mass curves spacetime**. Massive objects like planets and stars create distortions, bending the paths of objects and even electromagnetic waves like light. This curved geometry replaces Minkowski space with a more complex spacetime that adapts to the distribution of mass and energy. + +This concept of curved spacetime can be described mathematically by the **Schwarzschild solution**, a specific solution to Einstein’s field equations. It describes the geometry around a massive, non-rotating object, leading us to concepts like **black holes** and **event horizons**. + ## General Relativity: Gravity as Curvature > May the Schwarz(schild Solution) be with you [^4] From c3bfaece1ef231418d36de5f58dff5a162061ed8 Mon Sep 17 00:00:00 2001 From: jvivian Date: Wed, 6 Nov 2024 13:36:02 -0800 Subject: [PATCH 12/15] Add final sections - just need to do an editing pass! --- posts/classical-mechanics/imgs/eddy.png | Bin 0 -> 46947 bytes posts/classical-mechanics/imgs/krusty.png | Bin 0 -> 58529 bytes posts/classical-mechanics/imgs/schwarz.png | Bin 0 -> 58223 bytes posts/classical-mechanics/index.qmd | 191 +++++++++++++++++++-- 4 files changed, 173 insertions(+), 18 deletions(-) create mode 100644 posts/classical-mechanics/imgs/eddy.png create mode 100644 posts/classical-mechanics/imgs/krusty.png create mode 100644 posts/classical-mechanics/imgs/schwarz.png diff --git a/posts/classical-mechanics/imgs/eddy.png 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You other brothers can't deny._ [^6] [^6]: Sir Mix-a-Lot - I Like Big Butts -Minkowski space describes a flat spacetime, where the geometry remains consistent everywhere. But what happens when **mass** is introduced? Einstein’s theory of general relativity builds on his previous framework by realizing **mass curves spacetime**. Massive objects like planets and stars create distortions, bending the paths of objects and even electromagnetic waves like light. This curved geometry replaces Minkowski space with a more complex spacetime that adapts to the distribution of mass and energy. +Minkowski space describes a flat spacetime, where the geometry remains consistent everywhere. But what happens when **mass** is introduced? Einstein’s theory of general relativity builds on his previous framework by realizing **gravity is not a force but a consequence of spacetime curvature**. Massive objects like planets and stars create distortions, bending the paths of objects and even electromagnetic waves like light. This curved geometry replaces Minkowski space with a more complex spacetime that adapts to the distribution of mass and energy. This concept of curved spacetime can be described mathematically by the **Schwarzschild solution**, a specific solution to Einstein’s field equations. It describes the geometry around a massive, non-rotating object, leading us to concepts like **black holes** and **event horizons**. @@ -1228,26 +1234,175 @@ This concept of curved spacetime can be described mathematically by the **Schwar [^4]: Space Balls - 1987 -Goal: Introduce general relativity’s core idea that mass and energy curve spacetime, with a specific focus on the Schwarzschild solution. +:::{.callout-caution} +The math here is complicated and so things will be kept top-level. While Einstein's equation may not appear intimidating, it uses a myriad of reduced terms that stand for advanced concepts like the **Ricci Curvature Tensor** which the **Einstein Tensor** is based on. +::: + +Einstein’s theory of **general relativity** describes how **mass and energy curve spacetime**, creating the effect we perceive as gravity. The fundamental equation that captures this relationship is Einstein's **field equation**: + +$$ +G_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu} +$$ + +In simpler terms: + +- $G_{\mu \nu}$: This term represents **spacetime curvature**. It’s called the **Einstein tensor**, which describes how spacetime is curved at each point. Specifically, it involves complex mathematics that measure how spacetime “bends.” +- $T_{\mu \nu}$: This term represents **energy and momentum** in the universe. Called the **stress-energy tensor**, it includes mass, energy, pressure, and stress (forces) within spacetime. +- $\frac{8 \pi G}{c^4}$: **Einstein Graviational Constant** + +Conceptually, Einstein’s equation tells us that **mass and energy** (on the right) **determine the curvature of spacetime** (on the left). So, massive objects like stars or planets create a “dent” in spacetime and this curvature dictates how objects move around them. + +### The Schwarzschild Solution: A Simple Case of Curved Spacetime + +The **Schwarzschild solution** is a specific solution to Einstein’s field equations that applies to a **spherically symmetric, non-rotating massive object** (like an idealized black hole or planet). + +The **Schwarzschild metric** describes the geometry of spacetime around such an object and is given by: + +$$ +ds^2 = -\left(1 - \frac{2GM}{c^2 r}\right) c^2 dt^2 + \left(1 - \frac{2GM}{c^2 r}\right)^{-1} dr^2 + r^2 d\Omega^2 +$$ + +Broken down: + +- $ds^2$: This is the **spacetime interval** (or proper distance) between two events in spacetime, affected by the gravitational field. +- $\left(1 - \frac{2GM}{c^2 r}\right)$: This factor modifies both the time and space terms and depends on **mass $M$** of the object and the **distance $r$** from its center. +- $r$: The radial distance from the center of the massive object. +- $d\Omega^2$: This term represents the spherical coordinates (angles) around the object and isn’t directly relevant to time dilation or the event horizon. + +### Schwarzschild Radius and The Event Horizon + +[^7]: Medieval Proverb + +The **Schwarzschild radius** $R_s$ is a critical radius derived from this solution. It’s defined as: + +$$ +R_s = \frac{2GM}{c^2} +$$ + +- This radius represents the **event horizon** of a black hole—the boundary beyond which nothing, not even light, can escape. + +- When $r = R_s$, the term $\left(1 - \frac{2GM}{c^2 r}\right)$ becomes zero in the time component, making $ds^2 \to 0$ for any finite time interval $dt$. This means time essentially “stops” from the perspective of a distant observer watching an object fall toward the black hole. + +### Curvature & Coordinate Singularities + +There are two key types of singularities associated with black holes: + +- **Coordinate Singularity (Event Horizon)** + - At $r = R_s$, the Schwarzschild metric appears to “break down” (it becomes infinite or zero). But this isn’t a true physical singularity—it’s a **coordinate singularity**. + - This can be thought of as a mathematical artifact of the coordinate system. The event horizon can be crossed by an infalling object without encountering infinite curvature. + - Using alternative coordinates, such as **Kruskal-Szekeres coordinates**, removes this “singularity,” allowing for a smooth path through the event horizon. -Key Points: Briefly explain how Einstein’s field equations describe how mass curves spacetime. Describe the Schwarzschild solution as a specific, spherically symmetric solution that describes the geometry outside a massive, non-rotating object. Explain the Schwarzschild radius and its role in defining the event horizon. +- **Curvature Singularity (Central Singularity)** + - At $r = 0$, the curvature of spacetime becomes **infinite**. This is a **true physical singularity** where densities become infinitely large, and the laws of physics as we know them break down. - This singularity is located at the center of a black hole and is inescapable once an object crosses the event horizon. -Connection: Use this to introduce black holes and explain why crossing the event horizon changes the nature of time and space. +### Inevitability Across The Event Horizon +> All Roads Lead To Rome [^7] -## Black Holes: The Inevitable Exit from Our Universe -Goal: Define black holes, focusing on the event horizon and the singularity. +Once an object crosses the event horizon at $r = R_s$, the roles of time and space coordinates **effectively switch**. Inside the event horizon: -Key Points: Distinguish between the event horizon (boundary from which light cannot escape) and the singularity (where spacetime curvature becomes infinite). Describe how the event horizon represents a “point of no return”—once crossed, all paths lead to the singularity. +- The **radial coordinate $r$** behaves like **time**: all paths lead inevitably toward $r = 0$, the singularity, much like how all paths in normal time lead toward the future. +- The **time coordinate $t$** behaves more like a spatial coordinate, losing its usual meaning. Inside the horizon, movement “forward in time” is equivalent to moving closer to the singularity. -Connection: Lead into a discussion of Kruskal-Szekeres coordinates as a way to better understand spacetime inside the horizon. +This is why **crossing the event horizon is a point of no return**—just as you can’t stop moving forward in time, you can’t stop moving toward the singularity once you’re inside. + +![Visualization of Light Cones in Time as a Function of Schwarzschild Radius [^8]](imgs/schwarz.png) + +[^8]: https://www.johnboccio.com/ + +This coordinate flipping is a consequence of the Schwarzschild Coordinates used. If we use a slightly different reformulation known as the **Eddington-Finkelstein Coordinates** we see the same effect of moving towards the curvature singularity without the sign flipping: + +![Eddington-Finkelstein Coordinate Plot [^8]](imgs/eddy.png) + +### Summary + +The Schwarzschild solution reveals: + +- **Massive objects curve spacetime** creating gravitational effects. +- **The Schwarzschild radius** marks the event horizon beyond which time and space switch roles. +- **Coordinate singularities** (event horizons) differ from **curvature singularities** (central singularities). ## Kruskal-Szekeres Coordinates: Inside the Event Horizon -Goal: Use Kruskal coordinates to explain why the singularity lies in the future of any object crossing the event horizon. +**Kruskal coordinates** extend the **Schwarzschild solution** by providing a way to visualize black holes in a way that doesn’t have the coordinate singularity at the event horizon. This extension allows us to map out the **paths of objects** inside and outside the event horizon, showing how they approach the singularity. + +In Kruskal coordinates: + +- The **event horizon** is not a boundary in the traditional sense; rather, it’s a point where **space and time axes flip**. +- These coordinates allow for a smooth, continuous depiction of spacetime across the event horizon, highlighting the inevitability of the singularity for any object that crosses this boundary. + +![Kruskal–Szekeres coordinates [^8]](imgs/krusty.png) + +### The Flip: Space and Time Axes Inside the Event Horizon + +In the **Schwarzschild solution**, time $t$ and radial distance $r$ are the primary coordinates: + +- **Outside the event horizon**, $t$ acts as the time coordinate and $r$ as the spatial coordinate. +- **Inside the event horizon** (i.e., for $r < R_s$, the Schwarzschild radius), the coordinates flip in a very real sense: + - **Radial distance $r$** now acts like a **time coordinate**—all paths move inexorably toward decreasing $r$, regardless of any attempts to escape. + - **Time $t$** behaves more like a **spatial coordinate**, losing its usual meaning of “forward in time.” + +This flip in coordinates means that **Time flows toward the singularity** for any object inside the event horizon, making the singularity effectively an **inevitable future** rather than a point in space you could avoid. + + +### Kruskal Coordinates Eliminate The Coordinate Event Horizon Coordinate Singularity + +Kruskal coordinates $(T, X)$ replace $(t, r)$ to eliminate the coordinate singularity at $r = R_s$, making it clear that: + +- **The event horizon is just another part of spacetime** in Kruskal coordinates, without the artificial “infinite” boundary seen in Schwarzschild coordinates. +- **Inside the event horizon**, the coordinate transformation shows all future paths converge toward the singularity at $r = 0$, regardless of any efforts by the object. + +In Kruskal diagrams: + +- The **singularity** at $r = 0$ forms a “line” in the future, with all paths ending at this point. +- The **event horizon** is a 45° line, where light rays moving toward the horizon just manage to hover there, unable to escape. +- **Outside the event horizon**, paths can move freely, but inside, every direction (even those that feel “upward” or “outward”) ultimately leads to the singularity. + + +### Why the Singularity Is Inevitable in the Future + +Since time and space effectively flip inside the event horizon, the singularity acts as a future point in **temporal terms**: + +- No path—whether for light, particles, or any object—can avoid the singularity once it crosses the event horizon. +- This setup leads to the conclusion that **the singularity lies in the future of all objects that cross the event horizon**. There’s no direction to move that doesn’t lead to it, just as there’s no way to avoid moving forward in time outside a black hole. + +### Relativity Redfines Space, Time, and Causality + +Kruskal coordinates provide a profound visualization of how **general relativity** reshapes our understanding of spacetime: + +- **Space and time are not fixed** but depend on the gravitational context, and near a black hole, these roles are transformed. +- **Causality** inside the event horizon is altered: what we consider “future” becomes tied to an inevitable destination (the singularity). + +# Timeline of The Universe +> The End + +$Ga$ is **Giga-Annum** or 1 billion Earth years + +:::{.column-screen} +```{mermaid} +timeline + Big Bang : -13.8 Ga + First Atoms Form : -13.7 Ga + First Stars Form : -13.5 Ga + Milky Way Forms : -13.4 Ga + Sun and Solar System Form : -4.6 Ga + + Present Day : 0 Ga + + Milky Way and Andromeda Galaxy Collision : +4 Ga + Sun Expands into a Red Giant : +5 Ga + Sun Becomes a White Dwarf : +7.5 Ga + Stellar Formation Ceases in Most Galaxies : +100 Ga + Degenerate Era Begins : +1,000 Ga + Black Holes Begin Evaporating via Hawking Radiation : +10^67 Ga + Black Hole Era Ends : +10^100 Ga + Dark Era Begins (Only Subatomic Particles Remain) : +10^120 Ga +``` +::: -Key Points:Explain how Kruskal coordinates extend the Schwarzschild solution, allowing us to visualize the paths of objects inside the event horizon. Describe how, inside the event horizon, the time and space axes effectively “flip”—$t$ becomes spatial, and $x$ becomes temporal. Discuss the inevitability of the singularity: once inside, all directions in time lead to the singularity. +# Acknowledgements +Thank you for reading! I hope you found some parts as mind-blowingly bizarre as I did. -Connection: This section ties together how both special and general relativity redefine our notions of space, time, and causality. +Special thanks to my friend Devery for agreeing to look over the math for me and Sam and Kaia for letting me yap at them about this while I was working on it. # Glossary From 5b7235defd2463ca0d48420072325818e74e249f Mon Sep 17 00:00:00 2001 From: jvivian Date: Wed, 6 Nov 2024 13:41:46 -0800 Subject: [PATCH 13/15] Add rendered post for blog --- docs/index.html | 502 ++++- docs/listings.json | 1 + docs/posts/classical-mechanics/imgs/eddy.png | Bin 0 -> 46947 bytes .../posts/classical-mechanics/imgs/krusty.png | Bin 0 -> 58529 bytes .../classical-mechanics/imgs/schwarz.png | Bin 0 -> 58223 bytes docs/posts/classical-mechanics/index.html | 1873 +++++++++++++++++ .../figure-html/fig-ampere-output-1.png | Bin 0 -> 150048 bytes .../figure-html/fig-calculus-output-1.png | Bin 0 -> 109335 bytes ...orrected_mechanics_comparison-output-1.png | Bin 0 -> 206217 bytes .../figure-html/fig-faraday-output-1.png | Bin 0 -> 180135 bytes .../fig-gauss_law_electric-output-1.png | Bin 0 -> 332521 bytes ...itational_force_visualization-output-1.png | Bin 0 -> 72981 bytes .../fig-length-contraction-output-1.png | Bin 0 -> 86941 bytes .../figure-html/fig-magnetic-output-1.png | Bin 0 -> 588439 bytes .../fig-minkowski_light_cone-output-1.png | Bin 0 -> 101717 bytes .../figure-html/fig-pythag-output-1.png | Bin 0 -> 116869 bytes .../figure-html/fig-sho-output-1.png | Bin 0 -> 51176 bytes ...ng_weight_visualization-dffad-output-1.png | Bin 0 -> 102056 bytes docs/posts/classical-mechanics/styles.css | 22 + docs/search.json | 156 +- 20 files changed, 2433 insertions(+), 121 deletions(-) create mode 100644 docs/posts/classical-mechanics/imgs/eddy.png create mode 100644 docs/posts/classical-mechanics/imgs/krusty.png create mode 100644 docs/posts/classical-mechanics/imgs/schwarz.png create mode 100644 docs/posts/classical-mechanics/index.html create mode 100644 docs/posts/classical-mechanics/index_files/figure-html/fig-ampere-output-1.png create mode 100644 docs/posts/classical-mechanics/index_files/figure-html/fig-calculus-output-1.png create mode 100644 docs/posts/classical-mechanics/index_files/figure-html/fig-corrected_mechanics_comparison-output-1.png create mode 100644 docs/posts/classical-mechanics/index_files/figure-html/fig-faraday-output-1.png create mode 100644 docs/posts/classical-mechanics/index_files/figure-html/fig-gauss_law_electric-output-1.png create mode 100644 docs/posts/classical-mechanics/index_files/figure-html/fig-gravitational_force_visualization-output-1.png create mode 100644 docs/posts/classical-mechanics/index_files/figure-html/fig-length-contraction-output-1.png create mode 100644 docs/posts/classical-mechanics/index_files/figure-html/fig-magnetic-output-1.png create mode 100644 docs/posts/classical-mechanics/index_files/figure-html/fig-minkowski_light_cone-output-1.png create mode 100644 docs/posts/classical-mechanics/index_files/figure-html/fig-pythag-output-1.png create mode 100644 docs/posts/classical-mechanics/index_files/figure-html/fig-sho-output-1.png create mode 100644 docs/posts/classical-mechanics/index_files/figure-html/fig-spring_weight_visualization-dffad-output-1.png create mode 100644 docs/posts/classical-mechanics/styles.css diff --git a/docs/index.html b/docs/index.html index ce21eba..a47fad8 100644 --- a/docs/index.html +++ b/docs/index.html @@ -2,12 +2,12 @@ - + -blog - Blast Beats and Beep Boops +Blast Beats and Beep Boops – blog + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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Pythagorean Time Dilation

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A Summary of Classical Mechanics

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Math
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Physics
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Author
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John Vivian

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November 6, 2024

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A Journey Through (Space)Time

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I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me 1

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+Motivation +
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This overview is roughly structured by the concepts I found most profound while reading and researching topics in Sean Carroll’s book The Biggest Ideas in The Universe: Space, Time, and Motion.

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  • While velocity only has meaning in other reference frames, acceleration is absolute, and can be identified intriniscally without one (in most cases).
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  • The squared term in the inverse square law is derived from the spacial geometry we live in (three dimensions).
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  • Magnetism only exists as a relativistic effect when an object moves relative to an electric field, which helps explain why we have never observed a magnetic monopole.
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  • The path a particle or system takes is always the most “economical” route through spacetime via minimization/maximization of action—a scalar quantity that describes how the balance of kinetic versus potential energy of a physical system changes.
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  • Conservation laws arise directly from symmetries in the laws of physics. I.e. The invariance of a system’s laws relate directly to a given conservation law: time invariance \(\rightarrow\) energy conservation, spatial invariance \(\rightarrow\) momentum conservation, and rotational symmetry \(\rightarrow\) angular momentum conservation.
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  • The speed of light, \(c\), provides a conversion factor between time and space. Like energy \(\Leftrightarrow\) mass and electricity \(\Leftrightarrow\) magnetism, they are aspects of the same phenomena.
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  • Lorentz transformations, which describe effects like length contraction and time dilation, can be derived from the invariant spacetime interval—analagous to the Pythagorean theorem—in the geometry of Minkowski space.
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  • Schwartzchild’s solution to General Relativity contained a curious mathematical singularity which—originally thought to be a coordinate singularity—turned out to be a real curvature singularity in spacetime: the infamous black hole.
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I’m neither a mathematician or physicist but do my best to clarify the meaning of each variable and operator in the equations introduced.

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Absolute Power, Absolute Acceleration

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Imagine yourself floating in deep space, with only distant stars as a backdrop. Would you be able to tell if you were moving? Surprisingly, no. As Einstein demonstrated, velocity only has meaning relative to another reference frame. If two people drift past each other, neither can determine who is “stationary” or “moving” in any absolute sense—both could be moving, or one could be stationary. In empty space, without other points of reference, velocity becomes entirely relative.

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Oddly, acceleration is different. If you were suddenly accelerated in your spacesuit, you’d feel it immediately—your body would press against the suit, jerking you forward. Acceleration creates a force that acts on you directly, causing a force gradient you can feel. Unlike velocity, which requires no force to sustain, acceleration always involves a force acting on a mass.

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In fact, you can create your own “accelerometer” with something as simple as a spring and weight. Hang the weight from a spring, and if you accelerate left or right, the weight will stretch the spring in the opposite direction. This setup works because acceleration produces a noticeable force, while constant velocity does not.

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+Figure 1: Spring with a hanging weight, stretching left as the system accelerates to the right. +
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Back on Earth, even when we’re standing still, we feel the effect of acceleration toward the planet’s center, which we experience as weight. This downward pull is balanced by the normal force of the ground pushing back up against us, allowing us to experience acceleration even when we appear stationary.

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Physics equations are extraordinary because they describe universal relationships in the natural world. For instance, the momentum of an object is simply the product of its mass and velocity: \[p = mv\] This relationship applies universally to every object with mass and velocity. And I think that’s pretty neat 2. Notably, the equation shows that momentum is directly proportional to velocity for a constant mass—doubling an object’s velocity doubles its momentum.

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Kinematic Equations

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Moving Through Time: Universal Relationships of Motion

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These equations describe universal relationships about an object’s motion with constant acceleration—useful for predicting where or how fast something will be at a given time.

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\[v = v_0 + at\]

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Velocity increases linearly with time when acceleration is constant

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\[s = s_0 + v_0 t + \frac{1}{2} at^2\]

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Position grows with time, influenced by both your initial speed and the acceleration

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\[v^2 = v_0^2 + 2a(s - s_0)\]

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The relationship between velocity and distance, without needing time

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An interesting consequence to these equations is given position or velocity as a function of time, the equation will predict the function’s value for whatever \(t\) is. Meaning, these universal relationships do not operate as if the past (fixed from our perspective) is fundamentally different from the future (not-fixed, from our perspective).

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Calculus addresses two fundamental problems involving limits: rates of change and accumulation.

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Derivatives: The derivative represents the rate of change of a quantity. It answers questions like, “How fast is something changing at an exact moment?” Calculus solves this by finding \(\frac{dx}{dt}\) as both \(dx\) and \(dt\) approach zero—a challenge because dividing by zero is undefined. By using limits, calculus lets us precisely calculate this “instantaneous” rate without ever actually dividing by zero.

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Integrals: The integral represents the accumulation of quantities over an interval, like the area under a curve. To do this, calculus sums up infinitely many tiny pieces (Riemann squares) as their width approaches zero. This limit-based approach lets us find the exact area, even under curves that are difficult or impossible to sum otherwise.

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+Figure 2: Given some continuous function, the derivative is instantaneous rate of change at a given point. +
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For example, if we expand a small change in a function, such as \((x + h)^2 - x^2\), and take the limit as \(h \rightarrow 0\), we end up with \(2x\)—the derivative of \(x^2\). What’s really magical is the inverse of \(x^2\) is the integral. This makes sense if you consider the derivative gives the rate of change (e.g. velocity from a function of position) and integration adds the accumulated change to get the original value (e.g. total distance from a function of velocity).

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\[ \frac{dx}{dt} f(x) = \int f(v) \space dt \]

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The kinematic equations can be derived from the basic definition of calculus. Given a change in position over time is velocity: \[\frac{dx}{dt} = v\]

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we want to express \(v\) in terms of \(t\), since that also changes over time.

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\[v(t) = v_0 + at\]

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Reformulate velocity as the first time derivative of position

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\[\frac{dx}{dt} = v_0 + at\]

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Then integrate with respect to time

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\[x(t) = \int (v_0 + at) dt\]

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Solving for the integral gives us

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\[x(t) = v_0t + \frac{1}{2}at^2 + C\]

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Where C would represent whatever the initial position is, which we can call \(x_0\). This gives us

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\[x(t) = x_0 + v_0t + \frac{1}{2}at^2\]

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You can plug this equation into \(v = v_0 + at\) and do some math to get:

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\[v^2 = v_0^2 + 2a(x-x0)\]

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The Universal Law of Gravitation

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Not the Deepak Chopra variety

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Newton’s Law of Gravitation describes the gravitational force between two masses, expressing how every mass exerts an attractive force on every other mass in the universe:

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\[ +F = G\frac{m_1 m_2}{r^2} +\]

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where:

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+Figure 3: Force vectors (equal and opposite for each mass) generated as a result of their respective masses and distance. +
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Why the Inverse-Square Law? Why \(r^2\)?

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The inverse-square relationship, \(\frac{1}{r^2}\), reflects a fundamental property of how forces and fields spread out in three-dimensional space. Newton’s insight into this came from observing that gravitational and other forces, like light intensity, diminish in strength with the square of the distance.

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In three-dimensional space, when a force radiates outward from a point source (such as a mass), it spreads uniformly over the surface of an expanding sphere centered around that source. The surface area of a sphere grows proportionally to \(r^2\), where \(r\) is the radius of the sphere. Consequently, as the distance \(r\) increases, the same force is distributed over a larger area, causing the intensity at any given point on that surface to diminish in proportion to \(\frac{1}{r^2}\).

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Thus, the inverse-square law isn’t arbitrary but rather a natural consequence of geometry in three-dimensional space. If we lived in a universe with different spatial dimensions, the exponent would change accordingly. For example, in a two-dimensional plane, the force would weaken with \(\frac{1}{r}\), while in four dimensions, it would decay with \(\frac{1}{r^3}\).

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Newton’s Universal Law of Gravitation describes the generated by two masses. This equation also ties together the inverse square law (decay of intensity of effects such as light, radiation, or gravity).

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\[F = G\frac{m_1m_2}{r^2}\]

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Why is \(r\) squared? Why not quadrupled or halved? The inverse square relationship emerges as a property of living in three dimensions. The volume of a sphere increases with the square of the radius: \(4\pi r^2\).

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Maxwell’s Equations

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The Original Field Science

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Maxwell’s Equations elegantly describe the behavior of electric and magnetic fields, revealing that these two forces are not separate but deeply interconnected. They provide a unified framework for understanding how electric charges and currents create fields that interact, propagate, and influence each other, ultimately giving rise to light itself.

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Gauss’s Law (Electric Field)

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\[\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}\]

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The del operator (\(\nabla\)) is a vector operator that captures how quantities change in space, and it’s central to expressing Maxwell’s equations in a compact, powerful form.

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  1. Gradient (\(\nabla \phi\)): When applied to a scalar field \(\phi\), \(\nabla\) (del) gives the gradient, pointing in the direction of the greatest rate of change. For example, in an electric potential field \(\phi\), the gradient \(\vec{E} = -\nabla\phi\) tells us how the electric field varies in space.

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  2. Divergence (\(\nabla \cdot\)): When \(\nabla\) is applied to a vector field with a dot product (\(\nabla \cdot\)), it gives the divergence. Divergence measures the “outflow” or “inflow” of a field at a point:

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    • Gauss’s Law for Electricity \((\nabla·\vec{E} = \frac{ρ}{ε_0})\): This tells us that the electric field \(\vec{E}\) diverges from electric charges.
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    • Gauss’s Law for Magnetism \((\nabla·\vec{B} = 0)\): This shows that magnetic field lines have no divergence—they form closed loops, indicating no magnetic monopoles.
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  3. Curl (\(\nabla \times\)): When \(\nabla\) is applied to a vector field with a cross product (\(\nabla \times\)), it gives the curl, measuring the “rotation” or “circulation” of a field:

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    • Faraday’s Law \((\nabla×\vec{E} = -\frac{∂\vec{B}}{∂t})\): A changing magnetic field \(\vec{B}\) induces a circulating electric field \(\vec{E}\).
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    • Ampère’s Law (with Maxwell’s Correction) \((\nabla×\vec{B} = μ_0\vec{J} + μ_0ε_0\frac{∂\vec{E}}{∂t})\): A changing electric field \(\vec{E}\) or current density \(\vec{J}\) induces a circulating magnetic field \(\vec{B}\).
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Summary:

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The del operator (\(\nabla\)) is a vector operator that operates on a: - Gradient (\(\nabla\phi\)): scalar field, producing a vector field. - Divergence (\(\nabla\cdot\)): vector field, producing a scalar field. - Curl (\(\nabla\times\)): vector field, producing a vector field.

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Gauss’s Law describes the relationship between electric charges and the electric field they produce. Specifically, it states that the electric flux (the “flow” of the electric field) through a closed surface is proportional to the total charge enclosed within that surface. For example, the electric field around a point charge (like an electron) points away from the charge if it’s positive, and towards the charge if it’s negative. The strength of the field is directly related to the amount of charge.

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+Figure 4: Gauss’s Law for Electricity: Electric Field Lines Around Charges +
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Gauss’s Law for Magnetism

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\[\nabla \cdot \mathbf{B} = 0\]

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This law states that there are no magnetic monopoles — that is, no isolated north or south magnetic poles. Magnetic fields (\(\mathbf{B}\)) always form closed loops. Every magnetic field line that leaves a north pole must eventually return to a south pole.

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+Figure 5: Magnetic field lines around a magnetic point +
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Faraday’s Law of Induction

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\[\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\]

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Meaning: Faraday’s Law explains how a changing magnetic field creates an electric field. If you have a time-varying magnetic field (\(\mathbf{B}\)), it induces a circulating electric field (\(\mathbf{E}\)). This is the principle behind electromagnetic induction, which is used in transformers and electric generators. When a magnet is spun around near a coil of wire, the changing magnetic field induces an electric current in the wire, which we can then use as electrical power.

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  • \(\nabla \times \mathbf{E}\): This is the curl of the electric field \(\mathbf{E}\), which measures how much the electric field is “circulating” around a point.
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+Figure 6: Changing magnetic fields induce electric fields +
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Ampère’s Law (with Maxwell’s Correction)

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\[\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\]

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Ampère’s Law relates electric currents and changing electric fields to the magnetic field they produce. It says that magnetic fields can be generated in two ways:

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  • By electric currents (moving charges), as described by the first term, \(\mu_0 \mathbf{J}\).
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  • By changing electric fields, as described by the second term, \(\mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\), which is called the displacement current (introduced by Maxwell).
    • +
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Ampère’s Law with Maxwell’s correction tells us that currents (moving charges) and changing electric fields generate magnetic fields. Without the time-dependent term \(\frac{\partial \mathbf{E}}{\partial t}\), this would just be the original Ampère’s Law, which describes how an electric current creates a magnetic field (like in an electromagnet). Maxwell’s addition was crucial because it showed that even in the absence of a physical current, a changing electric field can still produce a magnetic field.

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+Figure 7: The opposite of Faraday’s Law - a time-varying electric field is the source of an induced magnetic field. +
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The Unification of Electromagnetism

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Maxwell’s equations, taken together, explain how electric and magnetic fields interact and propagate. If you combine Faraday’s Law and Ampère’s Law, you can derive the wave equation for both electric and magnetic fields, showing that disturbances in these fields propagate at the speed of light. Together, these two effects allow electromagnetic waves to propagate through space, with the electric and magnetic fields continuously regenerating each other.

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This is why light, radio waves, X-rays, and all other electromagnetic radiation are essentially electromagnetic waves: oscillating electric and magnetic fields that move through space, even in a vacuum.

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Lagrangian and Hamiltonian Mechanics

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Different ways to tell the same (Newtonian) story

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In Newtonian mechanics, where \(F = ma\), we focus on forces acting at a specific point in space and time to answer “What happens at the next \(t\)?” This approach helps us understand motion in terms of forces and accelerations.

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In contrast, mechanics based on action principles—Lagrangian and Hamiltonian mechanics—start from a different perspective, emphasizing the concept of action. Action represents an energy tradeoff between kinetic and potential energy, shaped by the system’s physical properties. Instead of predicting the immediate “next step,” these approaches answer questions about paths, e.g. which trajectory will efficiently guide a rocket to the moon?

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In lieu of forces, Lagrangian and Hamiltonian mechanics prioritize paths, energy, and phase space. Phase space, in particular, provides a powerful framework for visualizing the state of a system and plays a central role in fields like quantum mechanics and field theory.

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To see these differences in action, we can visualize a classical physical system—the simple harmonic oscillator—through the lenses of Newtonian, Lagrangian, and Hamiltonian mechanics.

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The Simple Harmonic Oscillator is a fundamental model of periodic motion, where an object experiences a restoring force proportional to its distance from an equilibrium position. This force always acts toward the equilibrium, driving the object back whenever it is displaced.

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+Figure 8: This mass-spring system illustrates the concept of a simple harmonic oscillator, where the restoring force is proportional to displacement. This same principle applies in other contexts such as pendulums, which exhibit harmonic motion at small angles. LC circuits, oscillate energy between electric and magnetic fields. This even extends to quantum harmonic oscillator models which models particles in a potential well. +
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+Figure 9: Comparison of Newtonian, Lagrangian, and Hamiltonian Mechanics for a simple harmonic oscillator +
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The Action Integral

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The profound economy of Nature

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The action integral \(S\) is the “score” of a given path taken by a system between two points in time.

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\[S[q(t)] = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt\]

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  • \(S[q(t)]\) is the action which depends on the path \(q(t)\)
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  • \(L(q, \dot{q}, t)\) is the Lagrangian, which is the difference between the kinetic energy \(T\) and potential energy \(V\)
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  • Note: Dot notation above a variable represents its first derivative. Two dots for the second derivative.
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\[L = T - V\]

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In essence, the action integral sums up the Lagrangian \(L\) over a specific path from \(t_1\) to \(t_2\). This principle of minimizing the action is known as the principle of least action, and it gives us the most efficient or “natural” path a system will take.

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The Euler-Lagrange Equation

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We’re still missing one component—a condition that allows us to minimize the Lagrangian. Thankfully someone way smarter than any of us worked out the rather elegant solution, the Euler-Lagrange equation:

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\[\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} = 0\]

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  • \(\frac{\partial L}{\partial \dot{q}}\): This is the partial derivative of the Lagrangian with respect to the velocity \(\dot{q}\). It measures how sensitive the Lagrangian is to changes in the system’s velocity.
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  • \(\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right)\): This term takes the time derivative of the above quantity, capturing how this sensitivity changes as the system evolves over time.
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  • \(\frac{\partial L}{\partial q}\): This is the partial derivative of the Lagrangian with respect to position \(q\). It measures how sensitive the Lagrangian is to changes in the system’s position.
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The Euler-Lagrange equation tells us that, for the actual path taken by the system, these two rates of change are equal and opposite.

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Intuition Behind the Equation

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The Euler-Lagrange equation is the mathematical expression of the principle of least action. It provides the conditions under which the action \(S\) is minimized.

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To build intuition around this concept, let’s imagine a ball rolling down a frictionless hill.

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When the ball starts at the top of the hill, it has maximum potential energy (due to height and the acceleration from gravity) and zero kinetic energy (energy due to motion). As it begins rolling down, gravity pulls it lower, decreasing its potential energy. But energy can’t just disappear—instead, it converts into kinetic energy, causing the ball to speed up.

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The ball follows a path where potential energy smoothly turns into kinetic energy as it descends. The principle of least action explains this smooth exchange by “favoring” paths where energy changes are gradual and balanced.

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When the ball reaches the bottom, it’s out of hill to roll down. Its potential energy is minimized, and all the energy it gained from descending has been converted into kinetic energy.

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This trade-off between potential and kinetic energy is natural because it minimizes abrupt changes in energy. A path where energy flowed in sharp bursts or stopped suddenly would require additional forces or disruptions. A smooth path minimizes these disturbances, making it the most efficient in terms of conserving energy while following the laws of physics.

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If this process sounds similar to heat diffusion, you’re doin good baby. Both are examples of the Universe’s tendency towards equilibrium while maintaining conservation.

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+TLDR +
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Any change in kinetic energy is exactly offset by an opposite change in potential energy, and vice versa.

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To see the action integral in action (hehe), let’s examine a simple harmonic oscillator, such as a mass attached to a spring and solve for the Lagrangian.

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Kinetic Energy \(T\)

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For an object of mass \(m\) moving with velocity \(\dot{q}\), the kinetic energy \(T\) is given by: \[T = \frac{1}{2} m \dot{q}^2\]

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Potential Energy \(V\)

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For a spring with spring constant \(k\), the potential energy \(V\) depends on the displacement \(q\) from equilibrium: \[V = \frac{1}{2} k q^2\]

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Lagrangian \(L = T - V\)

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Substituting the expressions for \(T\) and \(V\):

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\[L = T - V = \frac{1}{2} m \dot{q}^2 - \frac{1}{2} k q^2\]

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Action Integral

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To find the path \(q(t)\) that the system takes between times \(t_1\) and \(t_2\), we evaluate the action integral: \[S[q(t)] = \int_{t_1}^{t_2} \left( \frac{1}{2} m \dot{q}^2 - \frac{1}{2} k q^2 \right) \, dt\]

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Finding the Path (Euler-Lagrange Equation)

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Applying the Euler-Lagrange equation to the Lagrangian leads to the equation of motion: \[\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} = 0\]

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Solving this gives the differential equation for a simple harmonic oscillator: \[m \ddot{q} + k q = 0\]

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which has solutions in the form of sinusoidal functions, describing the oscillatory motion \(q(t)\) over time.

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Ok, ok sorry. I tried to pull a fast one. Solving the Lagrangian only gave us this differential equation—it didn’t directly give us the solution for \(q(t)\), which is the explicit path or trajectory of the system over time.

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We can rewrite our solution as

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\[\ddot{q} + \frac{k}{m}q=0\]

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Let \(\omega = \sqrt{\frac{k}{m}}\), where \(\omega\) is the angular frequency of the oscillator. Now the equation becomes: \[\ddot{q} + \omega^2 q = 0\]

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This equation has a general solution of the form: \[q(t) = A \cos(\omega t) + B \sin(\omega t)\] where \(A\) and \(B\) are constants determined by the initial conditions of the system (such as initial position and velocity).

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Oh my god fuck off already

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Interpretation

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This solution, \(q(t) = A \cos(\omega t) + B \sin(\omega t)\), is the explicit time-dependent path or trajectory that the oscillator follows. It describes the oscillatory motion in terms of time, with:

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  • \(\omega\): The frequency of oscillation, showing how quickly the system oscillates.
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  • \(A\) and \(B\): Constants that depend on the starting conditions (e.g., where and how fast the mass was initially moving).
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If you squint, you’ll see we’ve derived Hooke’s Law. Since \(\ddot{q} = a\) and \(F=ma\):

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\[ ma = F = - kq \]

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where \(k\) is the spring constant AKA restoring force over the distance (\(\frac{N}{m}\)).

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The Hamiltonian Reform(ul)ation

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Puttin’ big E first

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We spent a a bit of time on Lagrangian mechanics because it also requires introduces several other interesting concepts like the principle of least action. The good news is we’ve done a lot of the heavy lifting and there’s nothing too crazy about now thinking about Hamiltonian mechanics, where we shift from paths to energies.

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While the Lagrangian \(L = T - V\) is built around action, the Hamiltonian is a powerful reformulation that shifts the focus to energy. The Hamiltonian \(H\) typically represents the total energy of a system \(H = T + V\) (kinetic + potential energy) and is built to highlight conservation laws.

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Unlike the Lagrangian, the Hamiltonian only depends on the position \(q\) and momentum \(p\), but not directly on \(t\) itself. This approach introduces phase space (a space defined by positions and momenta), allowing viusalization of a system’s entire state at any moment. Shifting from position/velocity to position/momentum reveals insights about stability, resonance, and even chaotic behavior in systems.

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Reviewing Figure 9 now may be elucidating. First, the phase space for the simple harmonic oscillator shows energy is conserved (closed ellipse). Second, thinking back to our ball-on-a-hill example and starting from the left and rolling down hill to the right, would hit maximum momentum at the bottom of the hill (\(x=0\)) and as it oscillated forever back-and-forth would follow a clockwise path around the ellipse (rolling back to where we started in this coordinate system means negative velocity therefore negative momentum).

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Here we can shortcut since we already did the Lagrangian solution.

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\[L = T - V = \frac{1}{2} m \dot{q}^2 - \frac{1}{2} k q^2\]

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Define the Hamiltonian \(H = T + V\) in Terms of Position and Momentum

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To switch to the Hamiltonian formalism, we need to express the system in terms of position \(q\) and momentum \(p\), where momentum \(p\) is defined as: \[p = \frac{\partial L}{\partial \dot{q}} = m \dot{q}\]

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Now, let’s express the Hamiltonian \(H\), which for many systems is the total energy \(T + V\): \[H = T + V = \frac{p^2}{2m} + \frac{1}{2} k q^2\]

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So, the Hamiltonian for a simple harmonic oscillator is: \[H(q, p) = \frac{p^2}{2m} + \frac{1}{2} k q^2\]

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Interpret the Hamiltonian

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The Hamiltonian here represents the total energy of the oscillator, with:

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  • \(\frac{p^2}{2m}\): The kinetic energy in terms of momentum \(p\)
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  • \(\frac{1}{2} k q^2\): The potential energy stored in the spring
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Derive the Equations of Motion Using Hamilton’s Equations

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In Hamiltonian mechanics, we use Hamilton’s equations to describe the time evolution of the system:

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\[\dot{q} = \frac{\partial H}{\partial p} \quad \text{and} \quad \dot{p} = -\frac{\partial H}{\partial q}\]

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Applying these to our Hamiltonian: 1. Find \(\dot{q}\): \[\dot{q} = \frac{\partial H}{\partial p} = \frac{p}{m}\] This confirms our definition of momentum, \(p = m \dot{q}\).

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  1. Find \(\dot{p}\): \[\dot{p} = -\frac{\partial H}{\partial q} = -k q\] This gives us the familiar equation of motion for a simple harmonic oscillator: \[m \ddot{q} + k q = 0\]
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Visualizing in Phase Space

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The Hamiltonian approach naturally introduces phase space, where each point represents a state of the system with coordinates \((q, p)\). For a simple harmonic oscillator: - The system’s trajectory in phase space forms an ellipse. - This elliptical trajectory represents the conservation of total energy: as the oscillator moves, \(q\) and \(p\) oscillate back and forth, but the total energy (Hamiltonian) remains constant.

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This example highlights the power of the Hamiltonian formalism in tracking conserved quantities and understanding system behavior in phase space.

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Pythagoras in Minkowski Space

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Roads? Where we’re going, we don’t need roads 3

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Special Relativity and the Insanity of \(c\)

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At 16 years old Einstein had thoughts about what it would be like to chase down a beam of light coming out of a flashlight. Classically, if he moved fast enough he could coast alongside that wave of light, like when a train matches the speed of a passing bird.

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An unexpected consequence of the Michelson-Morley Experiment of 1887—while looking for a theoretical “luminiferous aether”—was that the inferometer detected light moving the same speed regardless of direction, suggesting it was the same speed regardless of how the Earth moved through space, therefore independent of the observer’s motion. This led the groundwork for Einstein’s theory.

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The Speed of Light, \(c\), and the Foundations of Special Relativity

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Einstein realized that, to be consistent with Maxwell’s equations for electromagnetism, light’s speed must remain constant across all frames of reference. This insight forced a rethinking of space and time. The concepts of absolute time (a universal “clock”) and absolute space (a universal grid) had to be discarded. Instead, space and time are intertwined in such a way that the speed of light remains constant—a revolutionary shift that led to the notion of spacetime.

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  1. Invariance of \(c\): The speed of light in a vacuum is the same for all observers, regardless of their motion. This means that whether you’re standing still, driving a car, or hurtling through space, \(c\) is always the same.
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  2. Space and Time Adapt: For this invariance to hold, space and time themselves must adapt to keep the speed of light constant. If you move faster, your time slows down and lengths contract relative to slower-moving observers.
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  3. Relativity of Simultaneity: Events that appear simultaneous to one observer may not be simultaneous to another moving observer. This idea challenges the intuitive notion of a shared “now” and suggests that time flows differently depending on motion. An event in one observer’s past can be in another’s future depending on how fast they are moving relative to one another.
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  4. More directly, we now know this also limits informations ability to propagate faster than speed of light. Quantum entanglement, counter-intuitively, does not violate this principle.
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Time is Just Another Coordinate

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In everyday life, we already treat time and space as coordinates, although we often don’t think of them this way. Imagine you’re meeting a friend at a café. You naturally specify a location (space coordinates) and a time (time coordinate). Without both, you might arrive at the café only to miss each other by an hour or by a block.

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Einstein’s insight was that this relationship goes much deeper: space and time are not separate entities but are interwoven. Just as you can’t meet your friend without specifying both space and time, the universe itself weaves these dimensions together in a way that affects how time flows and distances change.

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We can now measure something called the spacetime interval that takes both space and time into account by plotting space against time and exploring how \(c\) creates boundaries within this space and how spacetime intervals (measured as the hypotenuse on this graph) remain constant for all observers (and lead to the derivations of the Lorentz Transformations).

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Lost In (Minkowski) Space: The Geometry of Spacetime

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We can described the unified spacetime framework mathematically using a special kind of geometry called Minkowski space. In this space, all objects project a causal light cone forward along the time axis that restricts what other events in spacetime they can interact with. Photons, which are massless, propagate along the diagonals \(y = \pm \space x\).

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+Figure 10: Light Cone in Minkowski Space. A stationary observer moves vertically along a ‘World Line’ +
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The Spacetime Interval

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In Minkowski space, the fundamental unit of “distance” isn’t a simple measurement of space or time alone—it’s a combination of both, called the spacetime interval. For two events separated by a distance \(x\) and a time \(t\), the spacetime interval \(s\) is given by a modified form of the Pythagorean Theorem:

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\[s^2 = c^2 t^2 - x^2\]

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+Figure 11: Real Time in Minkowski space +
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This interval differs from the usual Euclidean distance formula.

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  • \(c\) is the speed of light, a constant that connects space and time.
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  • The interval \(s\) combines spatial separation (\(x\)) and time separation (\(t\)) but with a minus sign in front of \(x^2\), which makes the geometry of Minkowski space different from ordinary space.
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  • Note how we keep units consistent by multiplying \(t\) by \(c\). We’re not in Kansas anymore 4.
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Why This Interval Matters

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The spacetime interval \(s\) has an important property: it’s invariant for all observers, regardless of their relative motion. This invariance means that while two observers might disagree on measurements of \(x\) and \(t\), they will always agree on the value of \(s\). In other words, even though space and time are experienced differently depending on motion, spacetime itself has an unchanging “distance” between events.

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This interval \(s\) is also known as proper time when it represents the “real time” experienced by an object moving along a specific path in spacetime. Just as distance measures separation in space, the spacetime interval \(s\) measures separation in spacetime.

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One unintuitive concept about Minkowski space is: due to the negative sign in how the spacetime interval is calculated, a longer line does not correspond to more time experienced. In fact, just the opposite is true. Relative to a stationary reference view, the closer on object moves to the speed of light the less time it will experience (Time Dilation) and the smaller the object will appear (Length Contraction).

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To derive the time dilation and length contraction formulas from the spacetime interval, we can start with the invariant interval in special relativity. The spacetime interval \(s^2\) between two events in Minkowski space is given by:

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\[ +s^2 = c^2 \Delta t^2 - \Delta x^2 +\]

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where:

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  • \(c\) is the speed of light,
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  • \(\Delta t\) is the time difference between the two events in one frame,
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  • \(\Delta x\) is the spatial separation between the events in the same frame.
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This interval \(s^2\) is invariant (it remains the same) in all inertial reference frames and is the basis for both derivations.

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1. Deriving Time Dilation

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Consider two events that occur at the same location in a moving frame \(S'\) (e.g., a clock moving with speed \(v\) relative to an observer in frame \(S\)).

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Setup:

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  • In the moving frame \(S'\), the spatial separation \(\Delta x' = 0\) because both events happen at the same place.
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  • Let \(\Delta t'\) be the time interval between the two events in \(S'\) (the proper time).
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  • In the stationary frame \(S\), let the events be separated by a time interval \(\Delta t\) and a spatial distance \(\Delta x\).
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Interval in Frame \(S'\):

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Since \(\Delta x' = 0\), \[ +s^2 = c^2 \Delta t'^2 +\]

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Interval in Frame \(S\):

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In the stationary frame \(S\), the events are separated in both space and time: \[ +s^2 = c^2 \Delta t^2 - \Delta x^2 +\]

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Since \(s^2\) is invariant, we equate the two expressions: \[ +c^2 \Delta t'^2 = c^2 \Delta t^2 - \Delta x^2 +\]

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Now, using the relationship \(\Delta x = v \Delta t\) (where \(v\) is the relative velocity of the two frames), \[ +c^2 \Delta t'^2 = c^2 \Delta t^2 - (v \Delta t)^2 +\]

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Factoring out \(\Delta t^2\), \[ +c^2 \Delta t'^2 = \Delta t^2 (c^2 - v^2) +\]

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Dividing both sides by \(c^2\), \[ +\Delta t'^2 = \Delta t^2 \left( 1 - \frac{v^2}{c^2} \right) +\]

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Taking the square root, \[ +\Delta t' = \Delta t \sqrt{1 - \frac{v^2}{c^2}} +\]

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Since \(\Delta t'\) is the proper time (the time in the frame where the events happen at the same location), we rearrange this to express \(\Delta t\) in terms of \(\Delta t'\): \[ +\Delta t = \frac{\Delta t'}{\sqrt{1 - \frac{v^2}{c^2}}} +\]

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This is the time dilation formula. It shows that the time interval \(\Delta t\) in the stationary frame \(S\) (observed time) is longer than the time interval \(\Delta t'\) in the moving frame \(S'\) (proper time). In other words, a moving clock runs slower when observed from a stationary frame.

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2. Deriving Length Contraction

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Consider two points at rest in frame \(S\) that define the endpoints of a rod, with a length \(L\) measured in the frame \(S\).

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Setup:

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  • In the stationary frame \(S\), the endpoints of the rod are separated by \(\Delta x = L\).
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  • In the moving frame \(S'\), the endpoints are separated by \(\Delta x' = L'\), which we want to find.
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  • The events marking the locations of the endpoints are simultaneous in \(S\) (so \(\Delta t = 0\) in \(S\)).
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Interval in Frame \(S\):

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Since the events are simultaneous in \(S\), \[ +s^2 = -\Delta x^2 = -L^2 +\]

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Interval in Frame \(S'\):

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In the moving frame \(S'\), where the rod is moving with velocity \(v\), the spatial separation is \(\Delta x' = L'\), and there may be a nonzero time difference \(\Delta t'\) between the events.

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Using the invariant interval, \[ +s^2 = c^2 \Delta t'^2 - L'^2 +\]

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Equating the intervals, \[ +-L^2 = c^2 \Delta t'^2 - L'^2 +\]

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where \(L\) is the length of the rod in the stationary frame \(S\), and \(L'\) is the length measured in the moving frame \(S'\). To proceed, let’s set up the interval conditions a bit more carefully.

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Clarify the Setup

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In the stationary frame \(S\):

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  • The events marking the endpoints of the rod are simultaneous (\(\Delta t = 0\)).
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  • The length \(L\) is the proper length, as it’s measured in the frame where the rod is at rest.
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In the moving frame \(S'\):

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  • The endpoints are separated by \(\Delta x' = L'\).
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  • Since the events are not simultaneous in \(S'\), there may be a time difference \(\Delta t'\) between observing each endpoint in \(S'\).
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Interval Invariance Between Frames

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From the invariance of the spacetime interval, we have: \[ +s^2 = -L^2 = c^2 \Delta t'^2 - L'^2 +\]

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Finding \(L'\) in Terms of \(L\)

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Since we want to measure the length of the rod in \(S'\) (the moving frame), we must consider the relativity of simultaneity. For length contraction, we want to eliminate \(\Delta t'\) because in \(S'\) the length \(L'\) is measured at a single moment (i.e., events should be simultaneous in \(S'\) to define the length).

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To eliminate \(\Delta t'\), use the fact that \(\Delta t'\) is related to the motion of the rod. Given \(\Delta x = L\) in \(S\), and knowing that \(S'\) moves at velocity \(v\) relative to \(S\), the time interval in \(S'\) can be expressed as: \[ +\Delta t' = \frac{v \Delta x}{c^2} = \frac{v L}{c^2} +\]

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Substitute \(\Delta t' = \frac{v L}{c^2}\) back into the interval equation: \[ +-L^2 = c^2 \left( \frac{v L}{c^2} \right)^2 - L'^2 +\]

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Simplify: \[ +-L^2 = \frac{v^2 L^2}{c^2} - L'^2 +\]

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Rearrange to solve for \(L'^2\): \[ +L'^2 = L^2 \left(1 - \frac{v^2}{c^2}\right) +\]

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Taking the square root: \[ +L' = L \sqrt{1 - \frac{v^2}{c^2}} +\]

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Final Length Contraction Formula

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This is the length contraction formula: \[ +L' = L \sqrt{1 - \frac{v^2}{c^2}} +\]

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where:

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  • \(L\) is the proper length (length of the rod in the frame where it is at rest),
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  • \(L'\) is the contracted length as observed in the frame moving at speed \(v\) relative to the rod.
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+Figure 12: Example of how long a 1 meter rod would appear to an observer as they moved at different fractions of the speed of light +
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The Twin Paradox

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Now you might be feeling comfortable with the classic Twin Paradox where Alice and Bob age differently as one leaves the planet and returns going some speed close to \(c\). If you’re like me, you may suddenly think: “Wait, if every reference frame is relative, aren’t they both moving equally fast apart from one other, meaning they would both appear to experience less time?”

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The answer is yes. The only way the twin paradox actually makes sense is when you consider Bob has to stop his ship and turn around. If you imagine this path in Minkowski space, with Alice moving forward in time along the worldline axis and Bob moving away at some angle fractional trajectory of \(c\), then turning around and coming back, it becomes clear why this happens. The calculated spacetime interval for Bob will show that he has experienced less time than Alice due to his relative return speed. If Bob accelerated away very quickly then traveled back at a much lower speed, the age difference would be quite small compared to the standard scenario.

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Mass Consequences

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…And I cannot lie.
You other brothers can’t deny. 5

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Minkowski space describes a flat spacetime, where the geometry remains consistent everywhere. But what happens when mass is introduced? Einstein’s theory of general relativity builds on his previous framework by realizing gravity is not a force but a consequence of spacetime curvature. Massive objects like planets and stars create distortions, bending the paths of objects and even electromagnetic waves like light. This curved geometry replaces Minkowski space with a more complex spacetime that adapts to the distribution of mass and energy.

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This concept of curved spacetime can be described mathematically by the Schwarzschild solution, a specific solution to Einstein’s field equations. It describes the geometry around a massive, non-rotating object, leading us to concepts like black holes and event horizons.

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General Relativity: Gravity as Curvature

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May the Schwarz(schild Solution) be with you 6

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+Caution +
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The math here is complicated and so things will be kept top-level. While Einstein’s equation may not appear intimidating, it uses a myriad of reduced terms that stand for advanced concepts like the Ricci Curvature Tensor which the Einstein Tensor is based on.

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Einstein’s theory of general relativity describes how mass and energy curve spacetime, creating the effect we perceive as gravity. The fundamental equation that captures this relationship is Einstein’s field equation:

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\[ +G_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu} +\]

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In simpler terms:

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  • \(G_{\mu \nu}\): This term represents spacetime curvature. It’s called the Einstein tensor, which describes how spacetime is curved at each point. Specifically, it involves complex mathematics that measure how spacetime “bends.”
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  • \(T_{\mu \nu}\): This term represents energy and momentum in the universe. Called the stress-energy tensor, it includes mass, energy, pressure, and stress (forces) within spacetime.
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  • \(\frac{8 \pi G}{c^4}\): Einstein Graviational Constant
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Conceptually, Einstein’s equation tells us that mass and energy (on the right) determine the curvature of spacetime (on the left). So, massive objects like stars or planets create a “dent” in spacetime and this curvature dictates how objects move around them.

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The Schwarzschild Solution: A Simple Case of Curved Spacetime

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The Schwarzschild solution is a specific solution to Einstein’s field equations that applies to a spherically symmetric, non-rotating massive object (like an idealized black hole or planet).

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The Schwarzschild metric describes the geometry of spacetime around such an object and is given by:

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\[ +ds^2 = -\left(1 - \frac{2GM}{c^2 r}\right) c^2 dt^2 + \left(1 - \frac{2GM}{c^2 r}\right)^{-1} dr^2 + r^2 d\Omega^2 +\]

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Broken down:

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  • \(ds^2\): This is the spacetime interval (or proper distance) between two events in spacetime, affected by the gravitational field.
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  • \(\left(1 - \frac{2GM}{c^2 r}\right)\): This factor modifies both the time and space terms and depends on mass \(M\) of the object and the distance \(r\) from its center.
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  • \(r\): The radial distance from the center of the massive object.
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  • \(d\Omega^2\): This term represents the spherical coordinates (angles) around the object and isn’t directly relevant to time dilation or the event horizon.
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Schwarzschild Radius and The Event Horizon

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The Schwarzschild radius \(R_s\) is a critical radius derived from this solution. It’s defined as:

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\[ +R_s = \frac{2GM}{c^2} +\]

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  • This radius represents the event horizon of a black hole—the boundary beyond which nothing, not even light, can escape.

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  • When \(r = R_s\), the term \(\left(1 - \frac{2GM}{c^2 r}\right)\) becomes zero in the time component, making \(ds^2 \to 0\) for any finite time interval \(dt\). This means time essentially “stops” from the perspective of a distant observer watching an object fall toward the black hole.

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Curvature & Coordinate Singularities

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There are two key types of singularities associated with black holes:

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  • Coordinate Singularity (Event Horizon) +
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    • At \(r = R_s\), the Schwarzschild metric appears to “break down” (it becomes infinite or zero). But this isn’t a true physical singularity—it’s a coordinate singularity.
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    • This can be thought of as a mathematical artifact of the coordinate system. The event horizon can be crossed by an infalling object without encountering infinite curvature.
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    • Using alternative coordinates, such as Kruskal-Szekeres coordinates, removes this “singularity,” allowing for a smooth path through the event horizon.
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  • Curvature Singularity (Central Singularity) +
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    • At \(r = 0\), the curvature of spacetime becomes infinite. This is a true physical singularity where densities become infinitely large, and the laws of physics as we know them break down. - This singularity is located at the center of a black hole and is inescapable once an object crosses the event horizon.
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Inevitability Across The Event Horizon

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All Roads Lead To Rome 7

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Once an object crosses the event horizon at \(r = R_s\), the roles of time and space coordinates effectively switch. Inside the event horizon:

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  • The radial coordinate \(r\) behaves like time: all paths lead inevitably toward \(r = 0\), the singularity, much like how all paths in normal time lead toward the future.
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  • The time coordinate \(t\) behaves more like a spatial coordinate, losing its usual meaning. Inside the horizon, movement “forward in time” is equivalent to moving closer to the singularity.
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This is why crossing the event horizon is a point of no return—just as you can’t stop moving forward in time, you can’t stop moving toward the singularity once you’re inside.

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Visualization of Light Cones in Time as a Function of Schwarzschild Radius 8
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This coordinate flipping is a consequence of the Schwarzschild Coordinates used. If we use a slightly different reformulation known as the Eddington-Finkelstein Coordinates we see the same effect of moving towards the curvature singularity without the sign flipping:

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Eddington-Finkelstein Coordinate Plot 9
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Summary

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The Schwarzschild solution reveals:

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  • Massive objects curve spacetime creating gravitational effects.
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  • The Schwarzschild radius marks the event horizon beyond which time and space switch roles.
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  • Coordinate singularities (event horizons) differ from curvature singularities (central singularities).
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Kruskal-Szekeres Coordinates: Inside the Event Horizon

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Kruskal coordinates extend the Schwarzschild solution by providing a way to visualize black holes in a way that doesn’t have the coordinate singularity at the event horizon. This extension allows us to map out the paths of objects inside and outside the event horizon, showing how they approach the singularity.

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In Kruskal coordinates:

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  • The event horizon is not a boundary in the traditional sense; rather, it’s a point where space and time axes flip.
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  • These coordinates allow for a smooth, continuous depiction of spacetime across the event horizon, highlighting the inevitability of the singularity for any object that crosses this boundary.
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Kruskal–Szekeres coordinates 10
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The Flip: Space and Time Axes Inside the Event Horizon

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In the Schwarzschild solution, time \(t\) and radial distance \(r\) are the primary coordinates:

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  • Outside the event horizon, \(t\) acts as the time coordinate and \(r\) as the spatial coordinate.
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  • Inside the event horizon (i.e., for \(r < R_s\), the Schwarzschild radius), the coordinates flip in a very real sense: +
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    • Radial distance \(r\) now acts like a time coordinate—all paths move inexorably toward decreasing \(r\), regardless of any attempts to escape.
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    • Time \(t\) behaves more like a spatial coordinate, losing its usual meaning of “forward in time.”
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This flip in coordinates means that Time flows toward the singularity for any object inside the event horizon, making the singularity effectively an inevitable future rather than a point in space you could avoid.

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Kruskal Coordinates Eliminate The Coordinate Event Horizon Coordinate Singularity

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Kruskal coordinates \((T, X)\) replace \((t, r)\) to eliminate the coordinate singularity at \(r = R_s\), making it clear that:

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  • The event horizon is just another part of spacetime in Kruskal coordinates, without the artificial “infinite” boundary seen in Schwarzschild coordinates.
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  • Inside the event horizon, the coordinate transformation shows all future paths converge toward the singularity at \(r = 0\), regardless of any efforts by the object.
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In Kruskal diagrams:

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  • The singularity at \(r = 0\) forms a “line” in the future, with all paths ending at this point.
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  • The event horizon is a 45° line, where light rays moving toward the horizon just manage to hover there, unable to escape.
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  • Outside the event horizon, paths can move freely, but inside, every direction (even those that feel “upward” or “outward”) ultimately leads to the singularity.
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Why the Singularity Is Inevitable in the Future

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Since time and space effectively flip inside the event horizon, the singularity acts as a future point in temporal terms:

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  • No path—whether for light, particles, or any object—can avoid the singularity once it crosses the event horizon.
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  • This setup leads to the conclusion that the singularity lies in the future of all objects that cross the event horizon. There’s no direction to move that doesn’t lead to it, just as there’s no way to avoid moving forward in time outside a black hole.
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Relativity Redfines Space, Time, and Causality

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Kruskal coordinates provide a profound visualization of how general relativity reshapes our understanding of spacetime:

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  • Space and time are not fixed but depend on the gravitational context, and near a black hole, these roles are transformed.
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  • Causality inside the event horizon is altered: what we consider “future” becomes tied to an inevitable destination (the singularity).
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Timeline of The Universe

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\(Ga\) is Giga-Annum or 1 billion Earth years

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timeline
+    Big Bang : -13.8 Ga
+    First Atoms Form : -13.7 Ga
+    First Stars Form : -13.5 Ga
+    Milky Way Forms : -13.4 Ga
+    Sun and Solar System Form : -4.6 Ga
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+    Present Day : 0 Ga
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+    Milky Way and Andromeda Galaxy Collision : +4 Ga
+    Sun Expands into a Red Giant : +5 Ga
+    Sun Becomes a White Dwarf : +7.5 Ga
+    Stellar Formation Ceases in Most Galaxies : +100 Ga
+    Degenerate Era Begins : +1,000 Ga
+    Black Holes Begin Evaporating via Hawking Radiation : +10^67 Ga
+    Black Hole Era Ends : +10^100 Ga
+    Dark Era Begins (Only Subatomic Particles Remain) : +10^120 Ga
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Acknowledgements

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Thank you for reading! I hope you found some parts as mind-blowingly bizarre as I did.

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Special thanks to my friend Devery for agreeing to look over the math for me and Sam and Kaia for letting me yap at them about this while I was working on it.

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Glossary

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  • Momentum is the quantity of motion an object has—a measure of how hard it is to stop that object. Mathematically, it’s \(p = mv\), where \(p\) is momentum, \(m\) is mass, and \(v\) is velocity.
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  • Energy is the capacity to do work. It exists in various forms, such as kinetic energy, potential energy, thermal energy, etc. In mechanics, the most common types are: +
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    • Kinetic Energy is the energy an object has due to its motion: \(K = \frac{1}{2}mv^2\).
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    • Potential Energy is the energy an object has due to its position in a force field, such as gravitational potential energy \(U = mgh\) for an object in Earth’s gravitational field.
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  • Work is the process of energy transfer when a force is applied to an object and it moves: \(W = F \cdot d \cdot \cos(\theta)\), where \(F\) is the force, \(d\) is the displacement, and \(\theta\) is the angle between the force and displacement vectors.
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  • Newton’s Laws of Motion: +
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    • First Law (Inertia): An object will remain at rest or in uniform motion unless acted upon by an external force.
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    • Second Law: The acceleration of an object is proportional to the net force acting on it and inversely proportional to its mass: \(F = ma\).
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    • Third Law: For every action, there is an equal and opposite reaction.
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  • Maxwell’s Equations describe how electric and magnetic fields interact and propagate: +
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    • Gauss’s Law: The electric flux through a closed surface is proportional to the charge enclosed.
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    • Gauss’s Law for Magnetism: Magnetic monopoles do not exist; the magnetic flux through a closed surface is zero.
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    • Faraday’s Law of Induction: A changing magnetic field induces an electric field.
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    • Ampère’s Law (with Maxwell’s correction): Magnetic fields can be generated by electric currents and changing electric fields.
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  • Lagrangian Mechanics is a reformulation of classical mechanics based on the principle of least action. The Lagrangian \(L\) is defined as the difference between kinetic and potential energy: \(L = T - U\). It leads to the Euler-Lagrange equations, which describe the dynamics of a system.
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  • Hamiltonian Mechanics is another reformulation of classical mechanics, focused on energy conservation. The Hamiltonian \(H\) is the total energy of the system (kinetic plus potential energy): \(H = T + U\). It uses a set of equations called Hamilton’s equations to describe the evolution of a system over time.
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  • Minkowski Space is a four-dimensional space combining three-dimensional space with time into a single manifold, used in the theory of special relativity. It allows for a unified treatment of space and time with the spacetime interval: \(s^2 = (ct)^2 - x^2 - y^2 - z^2\), where \(c\) is the speed of light. Note how \(ct\) makes the units consistent.
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  • Einstein’s Theory of Special Relativity states that the laws of physics are the same in all inertial frames, and that the speed of light in a vacuum is constant. This leads to the famous equation \(E = mc^2\), which relates mass and energy.
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  • Einstein’s Theory of General Relativity generalizes special relativity to include gravity. It describes gravity not as a force but as the curvature of spacetime caused by mass and energy. Objects follow geodesics, the shortest paths, in this curved spacetime.
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Footnotes

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  1. Isaac Newton↩︎

    2. +

  2. Neature Walk: https://www.youtube.com/watch?v=Hm3JodBR-vs↩︎

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  3. Dr. Emmmet Brown - Back to the Future↩︎

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  4. Dorothy - The Wizard of Oz↩︎

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  5. Sir Mix-a-Lot - I Like Big Butts↩︎

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  6. Space Balls - 1987↩︎

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  7. Medieval Proverb↩︎

    8. +

  8. https://www.johnboccio.com/↩︎

    9. +

  9. https://www.johnboccio.com/↩︎

    10. +

  10. https://www.johnboccio.com/↩︎

    11. +
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Increases font size for edge labels */ +} + +.mermaid .label rect { + width: 200px; /* Adjusts width of nodes */ +} diff --git a/docs/search.json b/docs/search.json index d2f7d64..b759622 100644 --- a/docs/search.json +++ b/docs/search.json @@ -11,7 +11,7 @@ "href": "presentations/ideaya-presentation/index.html#nanopore-reread-project", "title": "Single-cell sequencing of the blood T cell repertoire", "section": "Nanopore reread project", - "text": "Nanopore reread project\n\nA: Substrate bound to the lipid bilayer via a cholesterol tag with a helicase (Hel308) bound and inactive at the other end. B: Electric potential causes the DNA duplex to unwind, leaving behind the cholesterol tether bound to the complement strand. C: Electric potential and the steady-state nature of the G-Quadruplex (GQ) causes it to unfold allowing the helicase to become enzymatically active in the 3’ -> 5’ direction. D: As Hel308 translocates the DNA back up through the pore, the GQ will refold allowing an additional enzyme to become bound. E: Continued translocation. F: More than 3-4 abasic residues causes Hel308 to dissociate from the substrate (9 are used in the substrate). The DNA strand is then pulled back down and the cycle is able to repeat itself once more." + "text": "Nanopore reread project\n\n\n\nA: Substrate bound to the lipid bilayer via a cholesterol tag with a helicase (Hel308) bound and inactive at the other end. B: Electric potential causes the DNA duplex to unwind, leaving behind the cholesterol tether bound to the complement strand. C: Electric potential and the steady-state nature of the G-Quadruplex (GQ) causes it to unfold allowing the helicase to become enzymatically active in the 3’ -> 5’ direction. D: As Hel308 translocates the DNA back up through the pore, the GQ will refold allowing an additional enzyme to become bound. E: Continued translocation. F: More than 3-4 abasic residues causes Hel308 to dissociate from the substrate (9 are used in the substrate). The DNA strand is then pulled back down and the cycle is able to repeat itself once more." }, { "objectID": "presentations/ideaya-presentation/index.html#computational-genomics-lab", @@ -88,7 +88,7 @@ "href": "presentations/ideaya-presentation/index.html#study-background", "title": "Single-cell sequencing of the blood T cell repertoire", "section": "Study Background", - "text": "Study Background\n\n\n\nHER2+ Breast Cancer\n\n\nHER2-positive breast cancer accounts for approximately 20% of breast cancer cases and is characterized by the overexpression of the human epidermal growth factor receptor 2 (HER2) gene 1\n\n\n\nAnti-HER2 treatments, such as trastuzumab, have drastically the management of HER2-positive breast cancer, turning a once difficult-to-treat cancer into one with much more favorable outcomes 2\n\n\n\n\n\nTumor-Infiltrating Lymphocytes\n\n\nAnti-tumor immunity plays a role in the efficacy of anti-HER2 treatment, as demonstrated by the association of TILs with prognosis and increased efficacy from treatment with trastuzumab and chemotherapy in HER2+ breast cancer 3\nImmunohistochemistry and RNA sequencing of breast tumor tissues has suggested that trastuzumab may recruit and activate anti-tumor T cells 4\n\n\n\n\nSlamon, D. J., et al. (2001). “Use of chemotherapy plus a monoclonal antibody against HER2 for metastatic breast cancer that overexpresses HER2.” New England Journal of Medicine, 344(11), 783-792.Piccart-Gebhart, M. J., et al. (2005). “Trastuzumab after adjuvant chemotherapy in HER2-positive breast cancer.” New England Journal of Medicine, 353(16), 1659-1672.Loi, S., et al. (2014). “Tumor infiltrating lymphocytes are prognostic in triple negative breast cancer and predictive for trastuzumab benefit in early breast cancer.” Ann Oncol, 25(8), 1544–1550.Hendricks, WPD., et al. (2019). “PD-1-Associated Gene Expression Signature of Neoadjuvant Trastuzumab-Treated Tumors Correlates with Patient Survival in HER2-Positive Breast Cancer.” Cancers, 11(10), 156." + "text": "Study Background\n\n\n\nHER2+ Breast Cancer\n\n\nHER2-positive breast cancer accounts for approximately 20% of breast cancer cases and is characterized by the overexpression of the human epidermal growth factor receptor 2 (HER2) gene 1\n\n\n\nAnti-HER2 treatments, such as trastuzumab, have drastically the management of HER2-positive breast cancer, turning a once difficult-to-treat cancer into one with much more favorable outcomes 2\n\n\n\n\n\nTumor-Infiltrating Lymphocytes\n\n\nAnti-tumor immunity plays a role in the efficacy of anti-HER2 treatment, as demonstrated by the association of TILs with prognosis and increased efficacy from treatment with trastuzumab and chemotherapy in HER2+ breast cancer 3\nImmunohistochemistry and RNA sequencing of breast tumor tissues has suggested that trastuzumab may recruit and activate anti-tumor T cells 4" }, { "objectID": "presentations/ideaya-presentation/index.html#study-design", @@ -116,7 +116,7 @@ "href": "presentations/ideaya-presentation/index.html#vdj-pipeline", "title": "Single-cell sequencing of the blood T cell repertoire", "section": "VDJ Pipeline", - "text": "VDJ Pipeline\n\nThe cellranger vdj pipeline can be used to analyze sequencing data produced from Chromium Next GEM Single Cell 5’ V(D)J libraries. It takes FASTQ files for V(D)J libraries and performs sequence assembly and paired clonotype calling. The pipeline uses the Chromium Cell Barcodes (also called 10x Barcodes) and UMIs to assemble V(D)J transcripts per cell." + "text": "VDJ Pipeline\n\n\n\nThe cellranger vdj pipeline can be used to analyze sequencing data produced from Chromium Next GEM Single Cell 5’ V(D)J libraries. It takes FASTQ files for V(D)J libraries and performs sequence assembly and paired clonotype calling. The pipeline uses the Chromium Cell Barcodes (also called 10x Barcodes) and UMIs to assemble V(D)J transcripts per cell." }, { "objectID": "presentations/ideaya-presentation/index.html#cite-seq", @@ -144,7 +144,7 @@ "href": "presentations/ideaya-presentation/index.html#batch-correction-and-dimensionality-reduction", "title": "Single-cell sequencing of the blood T cell repertoire", "section": "Batch Correction and Dimensionality Reduction", - "text": "Batch Correction and Dimensionality Reduction\n\nHarmony operates in embedding space and produces a corrected PCA embedding which is used downstream to calculate the neighborhood graph which in turn is used by UMAP for further dimnsionality reduction." + "text": "Batch Correction and Dimensionality Reduction\n\n\n\nHarmony operates in embedding space and produces a corrected PCA embedding which is used downstream to calculate the neighborhood graph which in turn is used by UMAP for further dimnsionality reduction." }, { "objectID": "presentations/ideaya-presentation/index.html#clustering-and-annotation", @@ -158,7 +158,7 @@ "href": "presentations/ideaya-presentation/index.html#patient-level-analysis", "title": "Single-cell sequencing of the blood T cell repertoire", "section": "Patient-level Analysis", - "text": "Patient-level Analysis\n\nLarge T cell clonal expansions were observed in pre- and on-treatment peripheral blood samples from two patients who exhibited a pathological complete response. A high abundance of activated CD8+ TEM/EM cells was seen in patients 1 and 5 (red arrows). The other three patients had partial clinical responses with residual disease at surgery. Patient 4 had an increase in activated CD4+ Tregs on-treatment (red circle)" + "text": "Patient-level Analysis\n\n\n\nLarge T cell clonal expansions were observed in pre- and on-treatment peripheral blood samples from two patients who exhibited a pathological complete response. A high abundance of activated CD8+ TEM/EM cells was seen in patients 1 and 5 (red arrows). The other three patients had partial clinical responses with residual disease at surgery. Patient 4 had an increase in activated CD4+ Tregs on-treatment (red circle)" }, { "objectID": "presentations/ideaya-presentation/index.html#variation-in-t-cell-populations-across-treatment-stages", @@ -172,14 +172,14 @@ "href": "presentations/ideaya-presentation/index.html#variation-in-t-cell-populations-across-treatment-stages-1", "title": "Single-cell sequencing of the blood T cell repertoire", "section": "Variation in T-Cell Populations Across Treatment Stages", - "text": "Variation in T-Cell Populations Across Treatment Stages\n\nA possible trend toward on-treatment increase in activated CD4+ Tregs was observed. Activated CD4+ Treg subset sizes are small and therefore estimation of abundance has high error bars." + "text": "Variation in T-Cell Populations Across Treatment Stages\n\n\n\nA possible trend toward on-treatment increase in activated CD4+ Tregs was observed. Activated CD4+ Treg subset sizes are small and therefore estimation of abundance has high error bars." }, { "objectID": "presentations/ideaya-presentation/index.html#summary", "href": "presentations/ideaya-presentation/index.html#summary", "title": "Single-cell sequencing of the blood T cell repertoire", "section": "Summary", - "text": "Summary\n\nSingle-cell mRNA sequencing enables high-resolution characterization of immune cell subsets and represents a promising approach to assess the immune response to anti-HER2 treatment and other cancer therapeutics\n\n\n\nIn this study, single-cell mRNA sequencing of peripheral CD3+ T-cells identified clonal expansions in activated CD8+ and CD4+ T cells in HER2+ breast cancer patients who had a pathological complete response with preoperative docetaxel, carboplatin, trastuzumab, and pertuzumab treatment\nThese data are consistent with the model that T-cells play a key role in anti-HER2-mediated tumor control and warrant further investigation in a larger sample population\n\n\n\n\nDue to the limited sample size (\\(n=5\\)), it was challenging to establish statistical correlations between complete pathological responses and the relative proportions of activated CD4/CD8 T-cell populations. Bayesian power analysis can be used to estimate the number of samples we would need to reliably quantify a given effect size." + "text": "Summary\n\nSingle-cell mRNA sequencing enables high-resolution characterization of immune cell subsets and represents a promising approach to assess the immune response to anti-HER2 treatment and other cancer therapeutics\n\n\n\nIn this study, single-cell mRNA sequencing of peripheral CD3+ T-cells identified clonal expansions in activated CD8+ and CD4+ T cells in HER2+ breast cancer patients who had a pathological complete response with preoperative docetaxel, carboplatin, trastuzumab, and pertuzumab treatment\nThese data are consistent with the model that T-cells play a key role in anti-HER2-mediated tumor control and warrant further investigation in a larger sample population\n\n\n. . .\n\nDue to the limited sample size (\\(n=5\\)), it was challenging to establish statistical correlations between complete pathological responses and the relative proportions of activated CD4/CD8 T-cell populations. Bayesian power analysis can be used to estimate the number of samples we would need to reliably quantify a given effect size." }, { "objectID": "posts/kd-art/index.html", @@ -697,7 +697,7 @@ "href": "index.html", "title": "Blast Beats and Beep Boops", "section": "", - "text": "Covid-19 Data-Rich Dynamic Factor Model\n\n\nOverview, Python Package, and Interactive Analysis\n\n\n\n\nMath\n\n\nModeling\n\n\nPython\n\n\n\n\n\n\n\n\n\n\n\nMar 9, 2024\n\n\nJohn Vivian, Aaron Cooke, Josh Fitzgerald\n\n\n\n\n\n\n \n\n\n\n\nInked Into Creation\n\n\nSettlement Intelligencia: Chapters I - XA Kingdom Death Tale\n\n\n\n\nWriting\n\n\nDigital Art\n\n\n\n\n\n\n\n\n\n\n\nFeb 26, 2024\n\n\nJohn Vivian\n\n\n\n\n\n\n \n\n\n\n\nKingdom Death Artwork\n\n\nPhysical Models as the Basis of Digital Art\n\n\n\n\nDigital Art\n\n\n\n\n\n\n\n\n\n\n\nNov 11, 2023\n\n\nJohn Vivian\n\n\n\n\n\n\n \n\n\n\n\nBrazilian Jiu Jitsu Concepts as Graphs\n\n\n\n\n\n\n\nBJJ\n\n\nGraphs\n\n\n\n\n\n\n\n\n\n\n\nAug 31, 2023\n\n\nJohn Vivian\n\n\n\n\n\n\n \n\n\n\n\nAwake\n\n\nA Kingdom Death Tale\n\n\n\n\nWriting\n\n\nDigital Art\n\n\n\n\n\n\n\n\n\n\n\nMay 2, 2023\n\n\nJohn Vivian\n\n\n\n\n\n\n \n\n\n\n\nHassberry-Pi: Part 1\n\n\nSetting up Home Assistant for Z-Wave, MQTT, Discord, and Raspberry Pi integration\n\n\n\n\nAutomation\n\n\nDocumentation\n\n\n\n\n\n\n\n\n\n\n\nMar 3, 2023\n\n\nJohn Vivian\n\n\n\n\n\n\nNo matching items\n\nCopyrightJohn Vivian - 2024 - All Rights Reserved" + "text": "Pythagorean Time Dilation\n\n\nA Summary of Classical Mechanics\n\n\n\nMath\n\n\nPhysics\n\n\n\n\n\n\n\n\n\nNov 6, 2024\n\n\nJohn Vivian\n\n\n\n\n\n\n\n\n\n\n\n\nCovid-19 Data-Rich Dynamic Factor Model\n\n\nOverview, Python Package, and Interactive Analysis\n\n\n\nMath\n\n\nModeling\n\n\nPython\n\n\n\n\n\n\n\n\n\nMar 9, 2024\n\n\nJohn Vivian, Aaron Cooke, Josh Fitzgerald\n\n\n\n\n\n\n\n\n\n\n\n\nInked Into Creation\n\n\nSettlement Intelligencia: Chapters I - XA Kingdom Death Tale\n\n\n\nWriting\n\n\nDigital Art\n\n\n\n\n\n\n\n\n\nFeb 26, 2024\n\n\nJohn Vivian\n\n\n\n\n\n\n\n\n\n\n\n\nKingdom Death Artwork\n\n\nPhysical Models as the Basis of Digital Art\n\n\n\nDigital Art\n\n\n\n\n\n\n\n\n\nNov 11, 2023\n\n\nJohn Vivian\n\n\n\n\n\n\n\n\n\n\n\n\nBrazilian Jiu Jitsu Concepts as Graphs\n\n\n\n\n\n\nBJJ\n\n\nGraphs\n\n\n\n\n\n\n\n\n\nAug 31, 2023\n\n\nJohn Vivian\n\n\n\n\n\n\n\n\n\n\n\n\nAwake\n\n\nA Kingdom Death Tale\n\n\n\nWriting\n\n\nDigital Art\n\n\n\n\n\n\n\n\n\nMay 2, 2023\n\n\nJohn Vivian\n\n\n\n\n\n\n\n\n\n\n\n\nHassberry-Pi: Part 1\n\n\nSetting up Home Assistant for Z-Wave, MQTT, Discord, and Raspberry Pi integration\n\n\n\nAutomation\n\n\nDocumentation\n\n\n\n\n\n\n\n\n\nMar 3, 2023\n\n\nJohn Vivian\n\n\n\n\n\n\nNo matching items\n\nCopyrightJohn Vivian - 2024 - All Rights Reserved" }, { "objectID": "posts/bjj-graphs/index.html", @@ -838,5 +838,145 @@ "title": "Single-cell sequencing of the blood T cell repertoire", "section": "Addendum: Bayesian Power Analysis", "text": "Addendum: Bayesian Power Analysis\n\n\n\n\n\n\nThis heatmap shows Bayesian power analysis outcomes, where the x-axis corresponds to sample size and the y-axis to effect size. Each cell shows the power (\\(p \\in [0, 1]\\)), which corresponds to the proportion of simulations detecting a significant difference. Color intensity denotes power levels, with darker colors signifying higher statistical power. Power analysis can help discern study feasibility as well as provide a sense of what effect size is detectable given sample size limitations.\n\n\n\n\nGeneral Steps\n\n\nDefine your prior beliefs about the parameters of interest\nSpecify the data generating process model. This model should represent the belief of how data arises within an experimental context\nDetermine the decision rule defining “significant” or “practically relevant.” A Bayesian approach is to specify a region of practical equivalence (ROPE) around a null value where effects are considered too small to be of interest\nSimulate data of varying sample sizes at differing effect sizes\nRun inference (e.g. MCMC) on each simulated dataset to estimate the posterior distributions\nCalculate power by determining how often the decision rule leads to a conclusion of practical significance by calculating the proportion of simulations where the posterior distribution of the effect size meets your criteria for significance (e.g., falls outside the ROPE)." + }, + { + "objectID": "posts/classical-mechanics/index.html", + "href": "posts/classical-mechanics/index.html", + "title": "Pythagorean Time Dilation", + "section": "", + "text": "I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me 1\n\n\n\n\n\n\n\nMotivation\n\n\n\nThis overview is roughly structured by the concepts I found most profound while reading and researching topics in Sean Carroll’s book The Biggest Ideas in The Universe: Space, Time, and Motion.\n\n\n\nWhile velocity only has meaning in other reference frames, acceleration is absolute, and can be identified intriniscally without one (in most cases).\nThe squared term in the inverse square law is derived from the spacial geometry we live in (three dimensions).\nMagnetism only exists as a relativistic effect when an object moves relative to an electric field, which helps explain why we have never observed a magnetic monopole.\nThe path a particle or system takes is always the most “economical” route through spacetime via minimization/maximization of action—a scalar quantity that describes how the balance of kinetic versus potential energy of a physical system changes.\nConservation laws arise directly from symmetries in the laws of physics. I.e. The invariance of a system’s laws relate directly to a given conservation law: time invariance \\(\\rightarrow\\) energy conservation, spatial invariance \\(\\rightarrow\\) momentum conservation, and rotational symmetry \\(\\rightarrow\\) angular momentum conservation.\nThe speed of light, \\(c\\), provides a conversion factor between time and space. Like energy \\(\\Leftrightarrow\\) mass and electricity \\(\\Leftrightarrow\\) magnetism, they are aspects of the same phenomena.\nLorentz transformations, which describe effects like length contraction and time dilation, can be derived from the invariant spacetime interval—analagous to the Pythagorean theorem—in the geometry of Minkowski space.\nSchwartzchild’s solution to General Relativity contained a curious mathematical singularity which—originally thought to be a coordinate singularity—turned out to be a real curvature singularity in spacetime: the infamous black hole.\n\n\n\n\n\n\n\nDisclaimer\n\n\n\nI’m neither a mathematician or physicist but do my best to clarify the meaning of each variable and operator in the equations introduced." + }, + { + "objectID": "posts/classical-mechanics/index.html#kinematic-equations", + "href": "posts/classical-mechanics/index.html#kinematic-equations", + "title": "Pythagorean Time Dilation", + "section": "Kinematic Equations", + "text": "Kinematic Equations\n\nMoving Through Time: Universal Relationships of Motion\n\nThese equations describe universal relationships about an object’s motion with constant acceleration—useful for predicting where or how fast something will be at a given time.\n\\[v = v_0 + at\\]\nVelocity increases linearly with time when acceleration is constant\n\\[s = s_0 + v_0 t + \\frac{1}{2} at^2\\]\nPosition grows with time, influenced by both your initial speed and the acceleration\n\\[v^2 = v_0^2 + 2a(s - s_0)\\]\nThe relationship between velocity and distance, without needing time\nAn interesting consequence to these equations is given position or velocity as a function of time, the equation will predict the function’s value for whatever \\(t\\) is. Meaning, these universal relationships do not operate as if the past (fixed from our perspective) is fundamentally different from the future (not-fixed, from our perspective).\n\n\n\n\n\n\nWhat’s the point of calculus\n\n\n\n\n\nCalculus addresses two fundamental problems involving limits: rates of change and accumulation.\nDerivatives: The derivative represents the rate of change of a quantity. It answers questions like, “How fast is something changing at an exact moment?” Calculus solves this by finding \\(\\frac{dx}{dt}\\) as both \\(dx\\) and \\(dt\\) approach zero—a challenge because dividing by zero is undefined. By using limits, calculus lets us precisely calculate this “instantaneous” rate without ever actually dividing by zero.\nIntegrals: The integral represents the accumulation of quantities over an interval, like the area under a curve. To do this, calculus sums up infinitely many tiny pieces (Riemann squares) as their width approaches zero. This limit-based approach lets us find the exact area, even under curves that are difficult or impossible to sum otherwise.\n\n\n\n\n\n\n\n\nFigure 2: Given some continuous function, the derivative is instantaneous rate of change at a given point.\n\n\n\n\n\nFor example, if we expand a small change in a function, such as \\((x + h)^2 - x^2\\), and take the limit as \\(h \\rightarrow 0\\), we end up with \\(2x\\)—the derivative of \\(x^2\\). What’s really magical is the inverse of \\(x^2\\) is the integral. This makes sense if you consider the derivative gives the rate of change (e.g. velocity from a function of position) and integration adds the accumulated change to get the original value (e.g. total distance from a function of velocity).\n\\[ \\frac{dx}{dt} f(x) = \\int f(v) \\space dt \\]\n\n\n\n\n\n\n\n\n\nKinematic Derivation\n\n\n\n\n\nThe kinematic equations can be derived from the basic definition of calculus. Given a change in position over time is velocity: \\[\\frac{dx}{dt} = v\\]\nwe want to express \\(v\\) in terms of \\(t\\), since that also changes over time.\n\\[v(t) = v_0 + at\\]\nReformulate velocity as the first time derivative of position\n\\[\\frac{dx}{dt} = v_0 + at\\]\nThen integrate with respect to time\n\\[x(t) = \\int (v_0 + at) dt\\]\nSolving for the integral gives us\n\\[x(t) = v_0t + \\frac{1}{2}at^2 + C\\]\nWhere C would represent whatever the initial position is, which we can call \\(x_0\\). This gives us\n\\[x(t) = x_0 + v_0t + \\frac{1}{2}at^2\\]\nYou can plug this equation into \\(v = v_0 + at\\) and do some math to get:\n\\[v^2 = v_0^2 + 2a(x-x0)\\]" + }, + { + "objectID": "posts/classical-mechanics/index.html#the-action-integral", + "href": "posts/classical-mechanics/index.html#the-action-integral", + "title": "Pythagorean Time Dilation", + "section": "The Action Integral", + "text": "The Action Integral\n\nThe profound economy of Nature\n\nThe action integral \\(S\\) is the “score” of a given path taken by a system between two points in time.\n\\[S[q(t)] = \\int_{t_1}^{t_2} L(q, \\dot{q}, t) \\, dt\\]\n\n\\(S[q(t)]\\) is the action which depends on the path \\(q(t)\\)\n\\(L(q, \\dot{q}, t)\\) is the Lagrangian, which is the difference between the kinetic energy \\(T\\) and potential energy \\(V\\)\nNote: Dot notation above a variable represents its first derivative. Two dots for the second derivative.\n\n\\[L = T - V\\]\nIn essence, the action integral sums up the Lagrangian \\(L\\) over a specific path from \\(t_1\\) to \\(t_2\\). This principle of minimizing the action is known as the principle of least action, and it gives us the most efficient or “natural” path a system will take.\n\nThe Euler-Lagrange Equation\nWe’re still missing one component—a condition that allows us to minimize the Lagrangian. Thankfully someone way smarter than any of us worked out the rather elegant solution, the Euler-Lagrange equation:\n\\[\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{q}} \\right) - \\frac{\\partial L}{\\partial q} = 0\\]\n\n\\(\\frac{\\partial L}{\\partial \\dot{q}}\\): This is the partial derivative of the Lagrangian with respect to the velocity \\(\\dot{q}\\). It measures how sensitive the Lagrangian is to changes in the system’s velocity.\n\\(\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{q}} \\right)\\): This term takes the time derivative of the above quantity, capturing how this sensitivity changes as the system evolves over time.\n\\(\\frac{\\partial L}{\\partial q}\\): This is the partial derivative of the Lagrangian with respect to position \\(q\\). It measures how sensitive the Lagrangian is to changes in the system’s position.\n\nThe Euler-Lagrange equation tells us that, for the actual path taken by the system, these two rates of change are equal and opposite.\n\nIntuition Behind the Equation\nThe Euler-Lagrange equation is the mathematical expression of the principle of least action. It provides the conditions under which the action \\(S\\) is minimized.\nTo build intuition around this concept, let’s imagine a ball rolling down a frictionless hill.\nWhen the ball starts at the top of the hill, it has maximum potential energy (due to height and the acceleration from gravity) and zero kinetic energy (energy due to motion). As it begins rolling down, gravity pulls it lower, decreasing its potential energy. But energy can’t just disappear—instead, it converts into kinetic energy, causing the ball to speed up.\nThe ball follows a path where potential energy smoothly turns into kinetic energy as it descends. The principle of least action explains this smooth exchange by “favoring” paths where energy changes are gradual and balanced.\nWhen the ball reaches the bottom, it’s out of hill to roll down. Its potential energy is minimized, and all the energy it gained from descending has been converted into kinetic energy.\nThis trade-off between potential and kinetic energy is natural because it minimizes abrupt changes in energy. A path where energy flowed in sharp bursts or stopped suddenly would require additional forces or disruptions. A smooth path minimizes these disturbances, making it the most efficient in terms of conserving energy while following the laws of physics.\nIf this process sounds similar to heat diffusion, you’re doin good baby. Both are examples of the Universe’s tendency towards equilibrium while maintaining conservation.\n\n\n\n\n\n\nTLDR\n\n\n\nAny change in kinetic energy is exactly offset by an opposite change in potential energy, and vice versa.\n\n\n\n\n\n\n\n\nThe Action of Simple Harmonic Motion\n\n\n\n\n\nTo see the action integral in action (hehe), let’s examine a simple harmonic oscillator, such as a mass attached to a spring and solve for the Lagrangian.\nKinetic Energy \\(T\\)\nFor an object of mass \\(m\\) moving with velocity \\(\\dot{q}\\), the kinetic energy \\(T\\) is given by: \\[T = \\frac{1}{2} m \\dot{q}^2\\]\nPotential Energy \\(V\\)\nFor a spring with spring constant \\(k\\), the potential energy \\(V\\) depends on the displacement \\(q\\) from equilibrium: \\[V = \\frac{1}{2} k q^2\\]\nLagrangian \\(L = T - V\\)\nSubstituting the expressions for \\(T\\) and \\(V\\):\n\\[L = T - V = \\frac{1}{2} m \\dot{q}^2 - \\frac{1}{2} k q^2\\]\nAction Integral\nTo find the path \\(q(t)\\) that the system takes between times \\(t_1\\) and \\(t_2\\), we evaluate the action integral: \\[S[q(t)] = \\int_{t_1}^{t_2} \\left( \\frac{1}{2} m \\dot{q}^2 - \\frac{1}{2} k q^2 \\right) \\, dt\\]\nFinding the Path (Euler-Lagrange Equation)\nApplying the Euler-Lagrange equation to the Lagrangian leads to the equation of motion: \\[\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{q}} \\right) - \\frac{\\partial L}{\\partial q} = 0\\]\nSolving this gives the differential equation for a simple harmonic oscillator: \\[m \\ddot{q} + k q = 0\\]\nwhich has solutions in the form of sinusoidal functions, describing the oscillatory motion \\(q(t)\\) over time.\n\n\n\n\n\n\nHey not so fast, where the hell is \\(q(t)\\)?\n\n\n\n\n\nOk, ok sorry. I tried to pull a fast one. Solving the Lagrangian only gave us this differential equation—it didn’t directly give us the solution for \\(q(t)\\), which is the explicit path or trajectory of the system over time.\nWe can rewrite our solution as\n\\[\\ddot{q} + \\frac{k}{m}q=0\\]\nLet \\(\\omega = \\sqrt{\\frac{k}{m}}\\), where \\(\\omega\\) is the angular frequency of the oscillator. Now the equation becomes: \\[\\ddot{q} + \\omega^2 q = 0\\]\nThis equation has a general solution of the form: \\[q(t) = A \\cos(\\omega t) + B \\sin(\\omega t)\\] where \\(A\\) and \\(B\\) are constants determined by the initial conditions of the system (such as initial position and velocity).\n\n\n\n\n\n\nBut wait, why is that the general solution?\n\n\n\n\n\nOh my god fuck off already\n\n\n\n\nInterpretation\nThis solution, \\(q(t) = A \\cos(\\omega t) + B \\sin(\\omega t)\\), is the explicit time-dependent path or trajectory that the oscillator follows. It describes the oscillatory motion in terms of time, with:\n\n\\(\\omega\\): The frequency of oscillation, showing how quickly the system oscillates.\n\\(A\\) and \\(B\\): Constants that depend on the starting conditions (e.g., where and how fast the mass was initially moving).\n\n\n\n\n\nIf you squint, you’ll see we’ve derived Hooke’s Law. Since \\(\\ddot{q} = a\\) and \\(F=ma\\):\n\\[ ma = F = - kq \\]\nwhere \\(k\\) is the spring constant AKA restoring force over the distance (\\(\\frac{N}{m}\\))." + }, + { + "objectID": "posts/classical-mechanics/index.html#footnotes", + "href": "posts/classical-mechanics/index.html#footnotes", + "title": "Pythagorean Time Dilation", + "section": "Footnotes", + "text": "Footnotes\n\n\nIsaac Newton↩︎\nNeature Walk: https://www.youtube.com/watch?v=Hm3JodBR-vs↩︎\nDr. Emmmet Brown - Back to the Future↩︎\nDorothy - The Wizard of Oz↩︎\nSir Mix-a-Lot - I Like Big Butts↩︎\nSpace Balls - 1987↩︎\nMedieval Proverb↩︎\nhttps://www.johnboccio.com/↩︎\nhttps://www.johnboccio.com/↩︎\nhttps://www.johnboccio.com/↩︎" + }, + { + "objectID": "presentations/ideaya-presentation/index.html#footnotes", + "href": "presentations/ideaya-presentation/index.html#footnotes", + "title": "Single-cell sequencing of the blood T cell repertoire", + "section": "Footnotes", + "text": "Footnotes\n\n\nSlamon, D. J., et al. (2001). “Use of chemotherapy plus a monoclonal antibody against HER2 for metastatic breast cancer that overexpresses HER2.” New England Journal of Medicine, 344(11), 783-792.↩︎\nPiccart-Gebhart, M. J., et al. (2005). “Trastuzumab after adjuvant chemotherapy in HER2-positive breast cancer.” New England Journal of Medicine, 353(16), 1659-1672.↩︎\nLoi, S., et al. (2014). “Tumor infiltrating lymphocytes are prognostic in triple negative breast cancer and predictive for trastuzumab benefit in early breast cancer.” Ann Oncol, 25(8), 1544–1550.↩︎\nHendricks, WPD., et al. (2019). “PD-1-Associated Gene Expression Signature of Neoadjuvant Trastuzumab-Treated Tumors Correlates with Patient Survival in HER2-Positive Breast Cancer.” Cancers, 11(10), 156.↩︎" + }, + { + "objectID": "posts/classical-mechanics/index.html#the-universal-law-of-gravitation", + "href": "posts/classical-mechanics/index.html#the-universal-law-of-gravitation", + "title": "Pythagorean Time Dilation", + "section": "The Universal Law of Gravitation", + "text": "The Universal Law of Gravitation\n\nNot the Deepak Chopra variety\n\nNewton’s Law of Gravitation describes the gravitational force between two masses, expressing how every mass exerts an attractive force on every other mass in the universe:\n\\[\nF = G\\frac{m_1 m_2}{r^2}\n\\]\nwhere:\n\n\\(F\\) is the gravitational force between the two masses.\n\\(G\\) is the gravitational constant, a fundamental constant that calibrates the strength of gravity.\n\\(m_1\\) and \\(m_2\\) are the masses of the two objects.\n\\(r\\) is the distance between the centers of the two masses.\n\n\n\n\n\n\n\n\n\nFigure 3: Force vectors (equal and opposite for each mass) generated as a result of their respective masses and distance.\n\n\n\n\n\n\nWhy the Inverse-Square Law? Why \\(r^2\\)?\nThe inverse-square relationship, \\(\\frac{1}{r^2}\\), reflects a fundamental property of how forces and fields spread out in three-dimensional space. Newton’s insight into this came from observing that gravitational and other forces, like light intensity, diminish in strength with the square of the distance.\nIn three-dimensional space, when a force radiates outward from a point source (such as a mass), it spreads uniformly over the surface of an expanding sphere centered around that source. The surface area of a sphere grows proportionally to \\(r^2\\), where \\(r\\) is the radius of the sphere. Consequently, as the distance \\(r\\) increases, the same force is distributed over a larger area, causing the intensity at any given point on that surface to diminish in proportion to \\(\\frac{1}{r^2}\\).\nThus, the inverse-square law isn’t arbitrary but rather a natural consequence of geometry in three-dimensional space. If we lived in a universe with different spatial dimensions, the exponent would change accordingly. For example, in a two-dimensional plane, the force would weaken with \\(\\frac{1}{r}\\), while in four dimensions, it would decay with \\(\\frac{1}{r^3}\\).\nNewton’s Universal Law of Gravitation describes the generated by two masses. This equation also ties together the inverse square law (decay of intensity of effects such as light, radiation, or gravity).\n\\[F = G\\frac{m_1m_2}{r^2}\\]\n\n\\(F\\) is the gravitational force between two masses.\n\\(G\\) is the gravitational constant.\n\\(m_1\\) and \\(m_2\\) are the masses of the two objects.\n\\(r\\) is the distance between the centers of the two masses.\n\nWhy is \\(r\\) squared? Why not quadrupled or halved? The inverse square relationship emerges as a property of living in three dimensions. The volume of a sphere increases with the square of the radius: \\(4\\pi r^2\\)." + }, + { + "objectID": "posts/classical-mechanics/index.html#gausss-law-electric-field", + "href": "posts/classical-mechanics/index.html#gausss-law-electric-field", + "title": "Pythagorean Time Dilation", + "section": "Gauss’s Law (Electric Field)", + "text": "Gauss’s Law (Electric Field)\n\\[\\nabla \\cdot \\mathbf{E} = \\frac{\\rho}{\\epsilon_0}\\]\n\n\n\n\n\n\nUnderstanding the Del Operator (\\(\\nabla\\)) in Maxwell’s Equations\n\n\n\n\n\nThe del operator (\\(\\nabla\\)) is a vector operator that captures how quantities change in space, and it’s central to expressing Maxwell’s equations in a compact, powerful form.\n\nGradient (\\(\\nabla \\phi\\)): When applied to a scalar field \\(\\phi\\), \\(\\nabla\\) (del) gives the gradient, pointing in the direction of the greatest rate of change. For example, in an electric potential field \\(\\phi\\), the gradient \\(\\vec{E} = -\\nabla\\phi\\) tells us how the electric field varies in space.\nDivergence (\\(\\nabla \\cdot\\)): When \\(\\nabla\\) is applied to a vector field with a dot product (\\(\\nabla \\cdot\\)), it gives the divergence. Divergence measures the “outflow” or “inflow” of a field at a point:\n\nGauss’s Law for Electricity \\((\\nabla·\\vec{E} = \\frac{ρ}{ε_0})\\): This tells us that the electric field \\(\\vec{E}\\) diverges from electric charges.\nGauss’s Law for Magnetism \\((\\nabla·\\vec{B} = 0)\\): This shows that magnetic field lines have no divergence—they form closed loops, indicating no magnetic monopoles.\n\nCurl (\\(\\nabla \\times\\)): When \\(\\nabla\\) is applied to a vector field with a cross product (\\(\\nabla \\times\\)), it gives the curl, measuring the “rotation” or “circulation” of a field:\n\nFaraday’s Law \\((\\nabla×\\vec{E} = -\\frac{∂\\vec{B}}{∂t})\\): A changing magnetic field \\(\\vec{B}\\) induces a circulating electric field \\(\\vec{E}\\).\nAmpère’s Law (with Maxwell’s Correction) \\((\\nabla×\\vec{B} = μ_0\\vec{J} + μ_0ε_0\\frac{∂\\vec{E}}{∂t})\\): A changing electric field \\(\\vec{E}\\) or current density \\(\\vec{J}\\) induces a circulating magnetic field \\(\\vec{B}\\).\n\n\n\nSummary:\nThe del operator (\\(\\nabla\\)) is a vector operator that operates on a: - Gradient (\\(\\nabla\\phi\\)): scalar field, producing a vector field. - Divergence (\\(\\nabla\\cdot\\)): vector field, producing a scalar field. - Curl (\\(\\nabla\\times\\)): vector field, producing a vector field.\n\n\n\n\nGauss’s Law describes the relationship between electric charges and the electric field they produce. Specifically, it states that the electric flux (the “flow” of the electric field) through a closed surface is proportional to the total charge enclosed within that surface. For example, the electric field around a point charge (like an electron) points away from the charge if it’s positive, and towards the charge if it’s negative. The strength of the field is directly related to the amount of charge.\n\n\\(\\nabla \\cdot \\mathbf{E}\\): This is the divergence of the electric field \\(\\mathbf{E}\\), which essentially measures how much the electric field is “spreading out” from a point.\n\\(\\rho\\): This represents the charge density (charge per unit volume). The more charge you have in a given region, the stronger the electric field diverges from that region.\n\\(\\epsilon_0\\): This is the permittivity of free space, a constant that determines how electric fields interact in a vacuum.\n\n\n\n\n\n\n\n\n\nFigure 4: Gauss’s Law for Electricity: Electric Field Lines Around Charges" + }, + { + "objectID": "posts/classical-mechanics/index.html#gausss-law-for-magnetism", + "href": "posts/classical-mechanics/index.html#gausss-law-for-magnetism", + "title": "Pythagorean Time Dilation", + "section": "Gauss’s Law for Magnetism", + "text": "Gauss’s Law for Magnetism\n\\[\\nabla \\cdot \\mathbf{B} = 0\\]\nThis law states that there are no magnetic monopoles — that is, no isolated north or south magnetic poles. Magnetic fields (\\(\\mathbf{B}\\)) always form closed loops. Every magnetic field line that leaves a north pole must eventually return to a south pole.\n\n\\(\\nabla \\cdot \\mathbf{B}\\): This is the divergence of the magnetic field \\(\\mathbf{B}\\), which measures how much the magnetic field is “spreading out” from a point.\nThe fact that it’s equal to zero means that the net magnetic “flux” through any closed surface is always zero, implying that magnetic field lines don’t begin or end at any point (unlike electric fields, which start or end on charges).\n\n\n\n\n\n\n\n\n\nFigure 5: Magnetic field lines around a magnetic point" + }, + { + "objectID": "posts/classical-mechanics/index.html#faradays-law-of-induction", + "href": "posts/classical-mechanics/index.html#faradays-law-of-induction", + "title": "Pythagorean Time Dilation", + "section": "Faraday’s Law of Induction", + "text": "Faraday’s Law of Induction\n\\[\\nabla \\times \\mathbf{E} = -\\frac{\\partial \\mathbf{B}}{\\partial t}\\]\nMeaning: Faraday’s Law explains how a changing magnetic field creates an electric field. If you have a time-varying magnetic field (\\(\\mathbf{B}\\)), it induces a circulating electric field (\\(\\mathbf{E}\\)). This is the principle behind electromagnetic induction, which is used in transformers and electric generators. When a magnet is spun around near a coil of wire, the changing magnetic field induces an electric current in the wire, which we can then use as electrical power.\n\n\\(\\nabla \\times \\mathbf{E}\\): This is the curl of the electric field \\(\\mathbf{E}\\), which measures how much the electric field is “circulating” around a point.\n\\(\\frac{\\partial \\mathbf{B}}{\\partial t}\\): This is the time rate of change of the magnetic field. It represents how the magnetic field is changing in time.\nThe negative sign indicates that the induced electric field opposes the change in the magnetic field (this is Lenz’s law).\n\n\n\n\n\n\n\n\n\nFigure 6: Changing magnetic fields induce electric fields" + }, + { + "objectID": "posts/classical-mechanics/index.html#ampères-law-with-maxwells-correction", + "href": "posts/classical-mechanics/index.html#ampères-law-with-maxwells-correction", + "title": "Pythagorean Time Dilation", + "section": "Ampère’s Law (with Maxwell’s Correction)", + "text": "Ampère’s Law (with Maxwell’s Correction)\n\\[\\nabla \\times \\mathbf{B} = \\mu_0 \\mathbf{J} + \\mu_0 \\epsilon_0 \\frac{\\partial \\mathbf{E}}{\\partial t}\\]\n\n\\(\\nabla \\times \\mathbf{B}\\): This is the curl of the magnetic field \\(\\mathbf{B}\\), which measures how much the magnetic field “circulates” around a point.\n\\(\\mathbf{J}\\): This is the current density (current per unit area).\n\\(\\frac{\\partial \\mathbf{E}}{\\partial t}\\): This represents the rate of change of the electric field in time.\n\nAmpère’s Law relates electric currents and changing electric fields to the magnetic field they produce. It says that magnetic fields can be generated in two ways:\n\nBy electric currents (moving charges), as described by the first term, \\(\\mu_0 \\mathbf{J}\\).\nBy changing electric fields, as described by the second term, \\(\\mu_0 \\epsilon_0 \\frac{\\partial \\mathbf{E}}{\\partial t}\\), which is called the displacement current (introduced by Maxwell).\n\nAmpère’s Law with Maxwell’s correction tells us that currents (moving charges) and changing electric fields generate magnetic fields. Without the time-dependent term \\(\\frac{\\partial \\mathbf{E}}{\\partial t}\\), this would just be the original Ampère’s Law, which describes how an electric current creates a magnetic field (like in an electromagnet). Maxwell’s addition was crucial because it showed that even in the absence of a physical current, a changing electric field can still produce a magnetic field.\n\n\n\n\n\n\n\n\nFigure 7: The opposite of Faraday’s Law - a time-varying electric field is the source of an induced magnetic field.\n\n\n\n\n\n\nThe Unification of Electromagnetism\nMaxwell’s equations, taken together, explain how electric and magnetic fields interact and propagate. If you combine Faraday’s Law and Ampère’s Law, you can derive the wave equation for both electric and magnetic fields, showing that disturbances in these fields propagate at the speed of light. Together, these two effects allow electromagnetic waves to propagate through space, with the electric and magnetic fields continuously regenerating each other.\nThis is why light, radio waves, X-rays, and all other electromagnetic radiation are essentially electromagnetic waves: oscillating electric and magnetic fields that move through space, even in a vacuum." + }, + { + "objectID": "posts/classical-mechanics/index.html#the-hamiltonian-reformulation", + "href": "posts/classical-mechanics/index.html#the-hamiltonian-reformulation", + "title": "Pythagorean Time Dilation", + "section": "The Hamiltonian Reform(ul)ation", + "text": "The Hamiltonian Reform(ul)ation\n\nPuttin’ big E first\n\nWe spent a a bit of time on Lagrangian mechanics because it also requires introduces several other interesting concepts like the principle of least action. The good news is we’ve done a lot of the heavy lifting and there’s nothing too crazy about now thinking about Hamiltonian mechanics, where we shift from paths to energies.\nWhile the Lagrangian \\(L = T - V\\) is built around action, the Hamiltonian is a powerful reformulation that shifts the focus to energy. The Hamiltonian \\(H\\) typically represents the total energy of a system \\(H = T + V\\) (kinetic + potential energy) and is built to highlight conservation laws.\nUnlike the Lagrangian, the Hamiltonian only depends on the position \\(q\\) and momentum \\(p\\), but not directly on \\(t\\) itself. This approach introduces phase space (a space defined by positions and momenta), allowing viusalization of a system’s entire state at any moment. Shifting from position/velocity to position/momentum reveals insights about stability, resonance, and even chaotic behavior in systems.\nReviewing Figure 9 now may be elucidating. First, the phase space for the simple harmonic oscillator shows energy is conserved (closed ellipse). Second, thinking back to our ball-on-a-hill example and starting from the left and rolling down hill to the right, would hit maximum momentum at the bottom of the hill (\\(x=0\\)) and as it oscillated forever back-and-forth would follow a clockwise path around the ellipse (rolling back to where we started in this coordinate system means negative velocity therefore negative momentum).\n\n\n\n\n\n\nSimple Harmonic Hamiltonian\n\n\n\n\n\n\nHere we can shortcut since we already did the Lagrangian solution.\n\n\\[L = T - V = \\frac{1}{2} m \\dot{q}^2 - \\frac{1}{2} k q^2\\]\nDefine the Hamiltonian \\(H = T + V\\) in Terms of Position and Momentum\nTo switch to the Hamiltonian formalism, we need to express the system in terms of position \\(q\\) and momentum \\(p\\), where momentum \\(p\\) is defined as: \\[p = \\frac{\\partial L}{\\partial \\dot{q}} = m \\dot{q}\\]\nNow, let’s express the Hamiltonian \\(H\\), which for many systems is the total energy \\(T + V\\): \\[H = T + V = \\frac{p^2}{2m} + \\frac{1}{2} k q^2\\]\nSo, the Hamiltonian for a simple harmonic oscillator is: \\[H(q, p) = \\frac{p^2}{2m} + \\frac{1}{2} k q^2\\]\nInterpret the Hamiltonian\nThe Hamiltonian here represents the total energy of the oscillator, with:\n\n\\(\\frac{p^2}{2m}\\): The kinetic energy in terms of momentum \\(p\\)\n\\(\\frac{1}{2} k q^2\\): The potential energy stored in the spring\n\nDerive the Equations of Motion Using Hamilton’s Equations\nIn Hamiltonian mechanics, we use Hamilton’s equations to describe the time evolution of the system:\n\\[\\dot{q} = \\frac{\\partial H}{\\partial p} \\quad \\text{and} \\quad \\dot{p} = -\\frac{\\partial H}{\\partial q}\\]\nApplying these to our Hamiltonian: 1. Find \\(\\dot{q}\\): \\[\\dot{q} = \\frac{\\partial H}{\\partial p} = \\frac{p}{m}\\] This confirms our definition of momentum, \\(p = m \\dot{q}\\).\n\nFind \\(\\dot{p}\\): \\[\\dot{p} = -\\frac{\\partial H}{\\partial q} = -k q\\] This gives us the familiar equation of motion for a simple harmonic oscillator: \\[m \\ddot{q} + k q = 0\\]\n\nVisualizing in Phase Space\nThe Hamiltonian approach naturally introduces phase space, where each point represents a state of the system with coordinates \\((q, p)\\). For a simple harmonic oscillator: - The system’s trajectory in phase space forms an ellipse. - This elliptical trajectory represents the conservation of total energy: as the oscillator moves, \\(q\\) and \\(p\\) oscillate back and forth, but the total energy (Hamiltonian) remains constant.\nThis example highlights the power of the Hamiltonian formalism in tracking conserved quantities and understanding system behavior in phase space." + }, + { + "objectID": "posts/classical-mechanics/index.html#special-relativity-and-insanity-of-c", + "href": "posts/classical-mechanics/index.html#special-relativity-and-insanity-of-c", + "title": "A Journey Through (Space)Time", + "section": "Special Relativity and Insanity of \\(c\\)", + "text": "Special Relativity and Insanity of \\(c\\)\nGoal: Introduce special relativity by focusing on why \\(c\\), the speed of light, is fundamental and invariant.\nKey Points: The postulate that \\(c\\) is the same for all observers leads to time dilation and length contraction. Explain how \\(c\\) creates a relationship between space and time, where distance can be “subtracted” from time, giving rise to the spacetime interval.\nConnection: This section sets up Minkowski space and real time, motivating the need for a geometric interpretation of spacetime." + }, + { + "objectID": "posts/classical-mechanics/index.html#minkowski-space", + "href": "posts/classical-mechanics/index.html#minkowski-space", + "title": "A Journey Through (Space)Time", + "section": "Minkowski Space", + "text": "Minkowski Space\nGoal: Introduce the geometry of Minkowski space and the concept of the spacetime interval.\nKey Points: Define Minkowski space and the spacetime interval: \\(s^2 = c^2t^2 - x^2\\). Explain that the interval \\(s\\) remains invariant for all observers and represents real time (proper time), which serves as the “distance” in this geometry.\nConnection: Introduce the idea that spacetime can be curved by mass, which lays the groundwork for general relativity and the Schwarzschild solution.\n\n\n\n\n\nFigure 10: Light Cone in Minkowski Space. A stationary observer moves vertically along a ‘World Line’\n\n\n\n\n\n\n\n\n\nFigure 11: Real Time in Minkowski space\n\n\n\n\n\n\n\n\n\nMinkowski Space: Worldlines for Stationary and Moving Observers" + }, + { + "objectID": "posts/classical-mechanics/index.html#general-relativity-gravity-as-curvature", + "href": "posts/classical-mechanics/index.html#general-relativity-gravity-as-curvature", + "title": "Pythagorean Time Dilation", + "section": "General Relativity: Gravity as Curvature", + "text": "General Relativity: Gravity as Curvature\n\nMay the Schwarz(schild Solution) be with you 6\n\n\n\n\n\n\n\nCaution\n\n\n\nThe math here is complicated and so things will be kept top-level. While Einstein’s equation may not appear intimidating, it uses a myriad of reduced terms that stand for advanced concepts like the Ricci Curvature Tensor which the Einstein Tensor is based on.\n\n\nEinstein’s theory of general relativity describes how mass and energy curve spacetime, creating the effect we perceive as gravity. The fundamental equation that captures this relationship is Einstein’s field equation:\n\\[\nG_{\\mu \\nu} = \\frac{8 \\pi G}{c^4} T_{\\mu \\nu}\n\\]\nIn simpler terms:\n\n\\(G_{\\mu \\nu}\\): This term represents spacetime curvature. It’s called the Einstein tensor, which describes how spacetime is curved at each point. Specifically, it involves complex mathematics that measure how spacetime “bends.”\n\\(T_{\\mu \\nu}\\): This term represents energy and momentum in the universe. Called the stress-energy tensor, it includes mass, energy, pressure, and stress (forces) within spacetime.\n\\(\\frac{8 \\pi G}{c^4}\\): Einstein Graviational Constant\n\nConceptually, Einstein’s equation tells us that mass and energy (on the right) determine the curvature of spacetime (on the left). So, massive objects like stars or planets create a “dent” in spacetime and this curvature dictates how objects move around them.\n\nThe Schwarzschild Solution: A Simple Case of Curved Spacetime\nThe Schwarzschild solution is a specific solution to Einstein’s field equations that applies to a spherically symmetric, non-rotating massive object (like an idealized black hole or planet).\nThe Schwarzschild metric describes the geometry of spacetime around such an object and is given by:\n\\[\nds^2 = -\\left(1 - \\frac{2GM}{c^2 r}\\right) c^2 dt^2 + \\left(1 - \\frac{2GM}{c^2 r}\\right)^{-1} dr^2 + r^2 d\\Omega^2\n\\]\nBroken down:\n\n\\(ds^2\\): This is the spacetime interval (or proper distance) between two events in spacetime, affected by the gravitational field.\n\\(\\left(1 - \\frac{2GM}{c^2 r}\\right)\\): This factor modifies both the time and space terms and depends on mass \\(M\\) of the object and the distance \\(r\\) from its center.\n\\(r\\): The radial distance from the center of the massive object.\n\\(d\\Omega^2\\): This term represents the spherical coordinates (angles) around the object and isn’t directly relevant to time dilation or the event horizon.\n\n\n\nSchwarzschild Radius and The Event Horizon\nThe Schwarzschild radius \\(R_s\\) is a critical radius derived from this solution. It’s defined as:\n\\[\nR_s = \\frac{2GM}{c^2}\n\\]\n\nThis radius represents the event horizon of a black hole—the boundary beyond which nothing, not even light, can escape.\nWhen \\(r = R_s\\), the term \\(\\left(1 - \\frac{2GM}{c^2 r}\\right)\\) becomes zero in the time component, making \\(ds^2 \\to 0\\) for any finite time interval \\(dt\\). This means time essentially “stops” from the perspective of a distant observer watching an object fall toward the black hole.\n\n\n\nCurvature & Coordinate Singularities\nThere are two key types of singularities associated with black holes:\n\nCoordinate Singularity (Event Horizon)\n\nAt \\(r = R_s\\), the Schwarzschild metric appears to “break down” (it becomes infinite or zero). But this isn’t a true physical singularity—it’s a coordinate singularity.\nThis can be thought of as a mathematical artifact of the coordinate system. The event horizon can be crossed by an infalling object without encountering infinite curvature.\nUsing alternative coordinates, such as Kruskal-Szekeres coordinates, removes this “singularity,” allowing for a smooth path through the event horizon.\n\nCurvature Singularity (Central Singularity)\n\nAt \\(r = 0\\), the curvature of spacetime becomes infinite. This is a true physical singularity where densities become infinitely large, and the laws of physics as we know them break down. - This singularity is located at the center of a black hole and is inescapable once an object crosses the event horizon.\n\n\n\n\nInevitability Across The Event Horizon\n\nAll Roads Lead To Rome 7\n\nOnce an object crosses the event horizon at \\(r = R_s\\), the roles of time and space coordinates effectively switch. Inside the event horizon:\n\nThe radial coordinate \\(r\\) behaves like time: all paths lead inevitably toward \\(r = 0\\), the singularity, much like how all paths in normal time lead toward the future.\nThe time coordinate \\(t\\) behaves more like a spatial coordinate, losing its usual meaning. Inside the horizon, movement “forward in time” is equivalent to moving closer to the singularity.\n\nThis is why crossing the event horizon is a point of no return—just as you can’t stop moving forward in time, you can’t stop moving toward the singularity once you’re inside.\n\n\n\nVisualization of Light Cones in Time as a Function of Schwarzschild Radius 8\n\n\nThis coordinate flipping is a consequence of the Schwarzschild Coordinates used. If we use a slightly different reformulation known as the Eddington-Finkelstein Coordinates we see the same effect of moving towards the curvature singularity without the sign flipping:\n\n\n\nEddington-Finkelstein Coordinate Plot 9\n\n\n\n\nSummary\nThe Schwarzschild solution reveals:\n\nMassive objects curve spacetime creating gravitational effects.\nThe Schwarzschild radius marks the event horizon beyond which time and space switch roles.\nCoordinate singularities (event horizons) differ from curvature singularities (central singularities)." + }, + { + "objectID": "posts/classical-mechanics/index.html#black-holes-the-inevitable-exit-from-our-universe", + "href": "posts/classical-mechanics/index.html#black-holes-the-inevitable-exit-from-our-universe", + "title": "Pythagorean Time Dilation", + "section": "Black Holes: The Inevitable Exit from Our Universe", + "text": "Black Holes: The Inevitable Exit from Our Universe\nGoal: Define black holes, focusing on the event horizon and the singularity.\nKey Points: Distinguish between the event horizon (boundary from which light cannot escape) and the singularity (where spacetime curvature becomes infinite). Describe how the event horizon represents a “point of no return”—once crossed, all paths lead to the singularity.\nConnection: Lead into a discussion of Kruskal-Szekeres coordinates as a way to better understand spacetime inside the horizon." + }, + { + "objectID": "posts/classical-mechanics/index.html#kruskal-szekeres-coordinates-inside-the-event-horizon", + "href": "posts/classical-mechanics/index.html#kruskal-szekeres-coordinates-inside-the-event-horizon", + "title": "Pythagorean Time Dilation", + "section": "Kruskal-Szekeres Coordinates: Inside the Event Horizon", + "text": "Kruskal-Szekeres Coordinates: Inside the Event Horizon\nKruskal coordinates extend the Schwarzschild solution by providing a way to visualize black holes in a way that doesn’t have the coordinate singularity at the event horizon. This extension allows us to map out the paths of objects inside and outside the event horizon, showing how they approach the singularity.\nIn Kruskal coordinates:\n\nThe event horizon is not a boundary in the traditional sense; rather, it’s a point where space and time axes flip.\nThese coordinates allow for a smooth, continuous depiction of spacetime across the event horizon, highlighting the inevitability of the singularity for any object that crosses this boundary.\n\n\n\n\nKruskal–Szekeres coordinates 10\n\n\n\nThe Flip: Space and Time Axes Inside the Event Horizon\nIn the Schwarzschild solution, time \\(t\\) and radial distance \\(r\\) are the primary coordinates:\n\nOutside the event horizon, \\(t\\) acts as the time coordinate and \\(r\\) as the spatial coordinate.\nInside the event horizon (i.e., for \\(r < R_s\\), the Schwarzschild radius), the coordinates flip in a very real sense:\n\nRadial distance \\(r\\) now acts like a time coordinate—all paths move inexorably toward decreasing \\(r\\), regardless of any attempts to escape.\nTime \\(t\\) behaves more like a spatial coordinate, losing its usual meaning of “forward in time.”\n\n\nThis flip in coordinates means that Time flows toward the singularity for any object inside the event horizon, making the singularity effectively an inevitable future rather than a point in space you could avoid.\n\n\nKruskal Coordinates Eliminate The Coordinate Event Horizon Coordinate Singularity\nKruskal coordinates \\((T, X)\\) replace \\((t, r)\\) to eliminate the coordinate singularity at \\(r = R_s\\), making it clear that:\n\nThe event horizon is just another part of spacetime in Kruskal coordinates, without the artificial “infinite” boundary seen in Schwarzschild coordinates.\nInside the event horizon, the coordinate transformation shows all future paths converge toward the singularity at \\(r = 0\\), regardless of any efforts by the object.\n\nIn Kruskal diagrams:\n\nThe singularity at \\(r = 0\\) forms a “line” in the future, with all paths ending at this point.\nThe event horizon is a 45° line, where light rays moving toward the horizon just manage to hover there, unable to escape.\nOutside the event horizon, paths can move freely, but inside, every direction (even those that feel “upward” or “outward”) ultimately leads to the singularity.\n\n\n\nWhy the Singularity Is Inevitable in the Future\nSince time and space effectively flip inside the event horizon, the singularity acts as a future point in temporal terms:\n\nNo path—whether for light, particles, or any object—can avoid the singularity once it crosses the event horizon.\nThis setup leads to the conclusion that the singularity lies in the future of all objects that cross the event horizon. There’s no direction to move that doesn’t lead to it, just as there’s no way to avoid moving forward in time outside a black hole.\n\n\n\nRelativity Redfines Space, Time, and Causality\nKruskal coordinates provide a profound visualization of how general relativity reshapes our understanding of spacetime:\n\nSpace and time are not fixed but depend on the gravitational context, and near a black hole, these roles are transformed.\nCausality inside the event horizon is altered: what we consider “future” becomes tied to an inevitable destination (the singularity)." + }, + { + "objectID": "posts/classical-mechanics/index.html#lost-in-minkowski-space", + "href": "posts/classical-mechanics/index.html#lost-in-minkowski-space", + "title": "Pythagorean Time Dilation", + "section": "Lost In (Minkowski) Space", + "text": "Lost In (Minkowski) Space\nHere’s an introduction to Minkowski space following those key points, setting up the concept of spacetime intervals and preparing for the idea of curved spacetime in general relativity.\n\n\nMinkowski Space: The Geometry of Spacetime\nIn everyday life, we think of space and time as separate. We measure distances in meters or miles and keep track of time with clocks. However, special relativity showed us that space and time are deeply connected, forming a unified framework called spacetime. To describe this framework mathematically, we use a special kind of geometry called Minkowski space.\n\n\nKey Concept: The Spacetime Interval\nIn Minkowski space, the fundamental unit of “distance” isn’t a simple measurement of space or time alone—it’s a combination of both, called the spacetime interval. For two events separated by a distance \\(x\\) and a time \\(t\\), the spacetime interval \\(s\\) is given by:\n[ s^2 = c^2 t^2 - x^2 ]\nThis interval differs from the usual Euclidean distance formula. Here: - \\(c\\) is the speed of light, a constant that connects space and time. - The interval \\(s\\) combines spatial separation (\\(x\\)) and time separation (\\(t\\)) but with a minus sign in front of \\(x^2\\), which makes the geometry of Minkowski space different from ordinary space.\n\n\nWhy This Interval Matters\nThe spacetime interval \\(s\\) has an important property: it’s invariant for all observers, regardless of their relative motion. This invariance means that while two observers might disagree on measurements of \\(x\\) and \\(t\\), they will always agree on the value of \\(s\\). In other words, even though space and time are experienced differently depending on motion, spacetime itself has an unchanging “distance” between events.\nThis interval \\(s\\) is also known as proper time when it represents the “real time” experienced by an object moving along a specific path in spacetime. Just as distance measures separation in space, the spacetime interval \\(s\\) measures separation in spacetime.\n\n\nVisualizing Minkowski Space\nIn Minkowski space, we can plot time and space as perpendicular axes, similar to a regular coordinate system, but with some key differences. Points (or “events”) in spacetime are located by both space and time coordinates. If we connect two events, the spacetime interval \\(s\\) between them becomes the “distance” in this spacetime geometry.\nThis interval sets limits on causality: events separated by a spacetime interval greater than zero (i.e., spacelike intervals) cannot influence each other, as no signal can travel faster than the speed of light. Light itself defines the boundary of possible influence, forming a “light cone” structure in this space.\n\n\nToward General Relativity: Curved Spacetime\nMinkowski space describes a flat spacetime, where the geometry remains consistent everywhere. But what happens when we introduce mass? Einstein’s theory of general relativity builds on this framework by showing that mass curves spacetime. Massive objects like planets and stars create distortions, bending the paths of objects and even light. This curved geometry replaces Minkowski space with a more complex spacetime that adapts to the distribution of mass and energy.\nThis concept of curved spacetime is described mathematically by the Schwarzschild solution, a specific solution to Einstein’s field equations. It describes the geometry around a massive, non-rotating object, leading us to concepts like black holes and event horizons.\nIn summary, Minkowski space provides the foundation for understanding spacetime as a unified geometry where space and time are linked through the spacetime interval. This flat spacetime model prepares us for the next step: seeing how spacetime itself bends in the presence of mass, laying the groundwork for the profound ideas of general relativity.\n\n\n\n\n\nFigure 10: Light Cone in Minkowski Space. A stationary observer moves vertically along a ‘World Line’\n\n\n\n\n\n\n\n\n\nFigure 11: Real Time in Minkowski space\n\n\n\n\n\n\n\n\n\nMinkowski Space: Worldlines for Stationary and Moving Observers" + }, + { + "objectID": "posts/classical-mechanics/index.html#special-relativity-and-the-insanity-of-c", + "href": "posts/classical-mechanics/index.html#special-relativity-and-the-insanity-of-c", + "title": "Pythagorean Time Dilation", + "section": "Special Relativity and the Insanity of \\(c\\)", + "text": "Special Relativity and the Insanity of \\(c\\)\nAt 16 years old Einstein had thoughts about what it would be like to chase down a beam of light coming out of a flashlight. Classically, if he moved fast enough he could coast alongside that wave of light, like when a train matches the speed of a passing bird.\nAn unexpected consequence of the Michelson-Morley Experiment of 1887—while looking for a theoretical “luminiferous aether”—was that the inferometer detected light moving the same speed regardless of direction, suggesting it was the same speed regardless of how the Earth moved through space, therefore independent of the observer’s motion. This led the groundwork for Einstein’s theory.\n\nThe Speed of Light, \\(c\\), and the Foundations of Special Relativity\nEinstein realized that, to be consistent with Maxwell’s equations for electromagnetism, light’s speed must remain constant across all frames of reference. This insight forced a rethinking of space and time. The concepts of absolute time (a universal “clock”) and absolute space (a universal grid) had to be discarded. Instead, space and time are intertwined in such a way that the speed of light remains constant—a revolutionary shift that led to the notion of spacetime.\n\nInvariance of \\(c\\): The speed of light in a vacuum is the same for all observers, regardless of their motion. This means that whether you’re standing still, driving a car, or hurtling through space, \\(c\\) is always the same.\nSpace and Time Adapt: For this invariance to hold, space and time themselves must adapt to keep the speed of light constant. If you move faster, your time slows down and lengths contract relative to slower-moving observers.\nRelativity of Simultaneity: Events that appear simultaneous to one observer may not be simultaneous to another moving observer. This idea challenges the intuitive notion of a shared “now” and suggests that time flows differently depending on motion. An event in one observer’s past can be in another’s future depending on how fast they are moving relative to one another.\nMore directly, we now know this also limits informations ability to propagate faster than speed of light. Quantum entanglement, counter-intuitively, does not violate this principle.\n\n\n\nTime is Just Another Coordinate\nIn everyday life, we already treat time and space as coordinates, although we often don’t think of them this way. Imagine you’re meeting a friend at a café. You naturally specify a location (space coordinates) and a time (time coordinate). Without both, you might arrive at the café only to miss each other by an hour or by a block.\nEinstein’s insight was that this relationship goes much deeper: space and time are not separate entities but are interwoven. Just as you can’t meet your friend without specifying both space and time, the universe itself weaves these dimensions together in a way that affects how time flows and distances change.\nWe can now measure something called the spacetime interval that takes both space and time into account by plotting space against time and exploring how \\(c\\) creates boundaries within this space and how spacetime intervals (measured as the hypotenuse on this graph) remain constant for all observers (and lead to the derivations of the Lorentz Transformations)." + }, + { + "objectID": "posts/classical-mechanics/index.html#lost-in-minkowski-space-the-geometry-of-spacetime", + "href": "posts/classical-mechanics/index.html#lost-in-minkowski-space-the-geometry-of-spacetime", + "title": "Pythagorean Time Dilation", + "section": "Lost In (Minkowski) Space: The Geometry of Spacetime", + "text": "Lost In (Minkowski) Space: The Geometry of Spacetime\nWe can described the unified spacetime framework mathematically using a special kind of geometry called Minkowski space. In this space, all objects project a causal light cone forward along the time axis that restricts what other events in spacetime they can interact with. Photons, which are massless, propagate along the diagonals \\(y = \\pm \\space x\\).\n\n\n\n\n\n\n\n\nFigure 10: Light Cone in Minkowski Space. A stationary observer moves vertically along a ‘World Line’\n\n\n\n\n\n\nThe Spacetime Interval\nIn Minkowski space, the fundamental unit of “distance” isn’t a simple measurement of space or time alone—it’s a combination of both, called the spacetime interval. For two events separated by a distance \\(x\\) and a time \\(t\\), the spacetime interval \\(s\\) is given by a modified form of the Pythagorean Theorem:\n\\[s^2 = c^2 t^2 - x^2\\]\n\n\n\n\n\n\n\n\nFigure 11: Real Time in Minkowski space\n\n\n\n\n\nThis interval differs from the usual Euclidean distance formula.\n\n\\(c\\) is the speed of light, a constant that connects space and time.\nThe interval \\(s\\) combines spatial separation (\\(x\\)) and time separation (\\(t\\)) but with a minus sign in front of \\(x^2\\), which makes the geometry of Minkowski space different from ordinary space.\nNote how we keep units consistent by multiplying \\(t\\) by \\(c\\). We’re not in Kansas anymore 4.\n\n\n\nWhy This Interval Matters\nThe spacetime interval \\(s\\) has an important property: it’s invariant for all observers, regardless of their relative motion. This invariance means that while two observers might disagree on measurements of \\(x\\) and \\(t\\), they will always agree on the value of \\(s\\). In other words, even though space and time are experienced differently depending on motion, spacetime itself has an unchanging “distance” between events.\nThis interval \\(s\\) is also known as proper time when it represents the “real time” experienced by an object moving along a specific path in spacetime. Just as distance measures separation in space, the spacetime interval \\(s\\) measures separation in spacetime.\nOne unintuitive concept about Minkowski space is: due to the negative sign in how the spacetime interval is calculated, a longer line does not correspond to more time experienced. In fact, just the opposite is true. Relative to a stationary reference view, the closer on object moves to the speed of light the less time it will experience (Time Dilation) and the smaller the object will appear (Length Contraction).\n\n\n\n\n\n\nDerivation of Lorentz Transformations from Spacetime Interval\n\n\n\n\n\nTo derive the time dilation and length contraction formulas from the spacetime interval, we can start with the invariant interval in special relativity. The spacetime interval \\(s^2\\) between two events in Minkowski space is given by:\n\\[\ns^2 = c^2 \\Delta t^2 - \\Delta x^2\n\\]\nwhere:\n\n\\(c\\) is the speed of light,\n\\(\\Delta t\\) is the time difference between the two events in one frame,\n\\(\\Delta x\\) is the spatial separation between the events in the same frame.\n\nThis interval \\(s^2\\) is invariant (it remains the same) in all inertial reference frames and is the basis for both derivations.\n\n\n1. Deriving Time Dilation\nConsider two events that occur at the same location in a moving frame \\(S'\\) (e.g., a clock moving with speed \\(v\\) relative to an observer in frame \\(S\\)).\n\nSetup:\n\nIn the moving frame \\(S'\\), the spatial separation \\(\\Delta x' = 0\\) because both events happen at the same place.\nLet \\(\\Delta t'\\) be the time interval between the two events in \\(S'\\) (the proper time).\nIn the stationary frame \\(S\\), let the events be separated by a time interval \\(\\Delta t\\) and a spatial distance \\(\\Delta x\\).\n\n\n\nInterval in Frame \\(S'\\):\nSince \\(\\Delta x' = 0\\), \\[\ns^2 = c^2 \\Delta t'^2\n\\]\n\n\nInterval in Frame \\(S\\):\nIn the stationary frame \\(S\\), the events are separated in both space and time: \\[\ns^2 = c^2 \\Delta t^2 - \\Delta x^2\n\\]\nSince \\(s^2\\) is invariant, we equate the two expressions: \\[\nc^2 \\Delta t'^2 = c^2 \\Delta t^2 - \\Delta x^2\n\\]\nNow, using the relationship \\(\\Delta x = v \\Delta t\\) (where \\(v\\) is the relative velocity of the two frames), \\[\nc^2 \\Delta t'^2 = c^2 \\Delta t^2 - (v \\Delta t)^2\n\\]\nFactoring out \\(\\Delta t^2\\), \\[\nc^2 \\Delta t'^2 = \\Delta t^2 (c^2 - v^2)\n\\]\nDividing both sides by \\(c^2\\), \\[\n\\Delta t'^2 = \\Delta t^2 \\left( 1 - \\frac{v^2}{c^2} \\right)\n\\]\nTaking the square root, \\[\n\\Delta t' = \\Delta t \\sqrt{1 - \\frac{v^2}{c^2}}\n\\]\nSince \\(\\Delta t'\\) is the proper time (the time in the frame where the events happen at the same location), we rearrange this to express \\(\\Delta t\\) in terms of \\(\\Delta t'\\): \\[\n\\Delta t = \\frac{\\Delta t'}{\\sqrt{1 - \\frac{v^2}{c^2}}}\n\\]\nThis is the time dilation formula. It shows that the time interval \\(\\Delta t\\) in the stationary frame \\(S\\) (observed time) is longer than the time interval \\(\\Delta t'\\) in the moving frame \\(S'\\) (proper time). In other words, a moving clock runs slower when observed from a stationary frame.\n\n\n\n\n2. Deriving Length Contraction\nConsider two points at rest in frame \\(S\\) that define the endpoints of a rod, with a length \\(L\\) measured in the frame \\(S\\).\n\nSetup:\n\nIn the stationary frame \\(S\\), the endpoints of the rod are separated by \\(\\Delta x = L\\).\nIn the moving frame \\(S'\\), the endpoints are separated by \\(\\Delta x' = L'\\), which we want to find.\nThe events marking the locations of the endpoints are simultaneous in \\(S\\) (so \\(\\Delta t = 0\\) in \\(S\\)).\n\n\n\nInterval in Frame \\(S\\):\nSince the events are simultaneous in \\(S\\), \\[\ns^2 = -\\Delta x^2 = -L^2\n\\]\n\n\nInterval in Frame \\(S'\\):\nIn the moving frame \\(S'\\), where the rod is moving with velocity \\(v\\), the spatial separation is \\(\\Delta x' = L'\\), and there may be a nonzero time difference \\(\\Delta t'\\) between the events.\nUsing the invariant interval, \\[\ns^2 = c^2 \\Delta t'^2 - L'^2\n\\]\nEquating the intervals, \\[\n-L^2 = c^2 \\Delta t'^2 - L'^2\n\\]\nwhere \\(L\\) is the length of the rod in the stationary frame \\(S\\), and \\(L'\\) is the length measured in the moving frame \\(S'\\). To proceed, let’s set up the interval conditions a bit more carefully.\n\n\nClarify the Setup\nIn the stationary frame \\(S\\):\n\nThe events marking the endpoints of the rod are simultaneous (\\(\\Delta t = 0\\)).\nThe length \\(L\\) is the proper length, as it’s measured in the frame where the rod is at rest.\n\nIn the moving frame \\(S'\\):\n\nThe endpoints are separated by \\(\\Delta x' = L'\\).\nSince the events are not simultaneous in \\(S'\\), there may be a time difference \\(\\Delta t'\\) between observing each endpoint in \\(S'\\).\n\n\n\nInterval Invariance Between Frames\nFrom the invariance of the spacetime interval, we have: \\[\ns^2 = -L^2 = c^2 \\Delta t'^2 - L'^2\n\\]\n\n\nFinding \\(L'\\) in Terms of \\(L\\)\nSince we want to measure the length of the rod in \\(S'\\) (the moving frame), we must consider the relativity of simultaneity. For length contraction, we want to eliminate \\(\\Delta t'\\) because in \\(S'\\) the length \\(L'\\) is measured at a single moment (i.e., events should be simultaneous in \\(S'\\) to define the length).\nTo eliminate \\(\\Delta t'\\), use the fact that \\(\\Delta t'\\) is related to the motion of the rod. Given \\(\\Delta x = L\\) in \\(S\\), and knowing that \\(S'\\) moves at velocity \\(v\\) relative to \\(S\\), the time interval in \\(S'\\) can be expressed as: \\[\n\\Delta t' = \\frac{v \\Delta x}{c^2} = \\frac{v L}{c^2}\n\\]\nSubstitute \\(\\Delta t' = \\frac{v L}{c^2}\\) back into the interval equation: \\[\n-L^2 = c^2 \\left( \\frac{v L}{c^2} \\right)^2 - L'^2\n\\]\nSimplify: \\[\n-L^2 = \\frac{v^2 L^2}{c^2} - L'^2\n\\]\nRearrange to solve for \\(L'^2\\): \\[\nL'^2 = L^2 \\left(1 - \\frac{v^2}{c^2}\\right)\n\\]\nTaking the square root: \\[\nL' = L \\sqrt{1 - \\frac{v^2}{c^2}}\n\\]\n\n\n\nFinal Length Contraction Formula\nThis is the length contraction formula: \\[\nL' = L \\sqrt{1 - \\frac{v^2}{c^2}}\n\\]\nwhere:\n\n\\(L\\) is the proper length (length of the rod in the frame where it is at rest),\n\\(L'\\) is the contracted length as observed in the frame moving at speed \\(v\\) relative to the rod.\n\n\n\n\n\n\n\n\n\n\n\n\n\nFigure 12: Example of how long a 1 meter rod would appear to an observer as they moved at different fractions of the speed of light\n\n\n\n\n\n\n\nThe Twin Paradox\nNow you might be feeling comfortable with the classic Twin Paradox where Alice and Bob age differently as one leaves the planet and returns going some speed close to \\(c\\). If you’re like me, you may suddenly think: “Wait, if every reference frame is relative, aren’t they both moving equally fast apart from one other, meaning they would both appear to experience less time?”\nThe answer is yes. The only way the twin paradox actually makes sense is when you consider Bob has to stop his ship and turn around. If you imagine this path in Minkowski space, with Alice moving forward in time along the worldline axis and Bob moving away at some angle fractional trajectory of \\(c\\), then turning around and coming back, it becomes clear why this happens. The calculated spacetime interval for Bob will show that he has experienced less time than Alice due to his relative return speed. If Bob accelerated away very quickly then traveled back at a much lower speed, the age difference would be quite small compared to the standard scenario.\n\n\nMass Consequences\n\n…And I cannot lie. You other brothers can’t deny. 5\n\nMinkowski space describes a flat spacetime, where the geometry remains consistent everywhere. But what happens when mass is introduced? Einstein’s theory of general relativity builds on his previous framework by realizing gravity is not a force but a consequence of spacetime curvature. Massive objects like planets and stars create distortions, bending the paths of objects and even electromagnetic waves like light. This curved geometry replaces Minkowski space with a more complex spacetime that adapts to the distribution of mass and energy.\nThis concept of curved spacetime can be described mathematically by the Schwarzschild solution, a specific solution to Einstein’s field equations. It describes the geometry around a massive, non-rotating object, leading us to concepts like black holes and event horizons." + }, + { + "objectID": "posts/classical-mechanics/index.html#black-holes-the-inevitable-exit", + "href": "posts/classical-mechanics/index.html#black-holes-the-inevitable-exit", + "title": "Pythagorean Time Dilation", + "section": "Black Holes: The Inevitable Exit", + "text": "Black Holes: The Inevitable Exit\n\nAll Roads Lead To Rome 7\n\nGoal: Define black holes, focusing on the event horizon and the singularity.\nKey Points: Distinguish between the event horizon (boundary from which light cannot escape) and the singularity (where spacetime curvature becomes infinite). Describe how the event horizon represents a “point of no return”—once crossed, all paths lead to the singularity.\nConnection: Lead into a discussion of Kruskal-Szekeres coordinates as a way to better understand spacetime inside the horizon." } ] \ No newline at end of file From c8f6d34d594c292906b9b3b53e38f6d1826ff548 Mon Sep 17 00:00:00 2001 From: jvivian Date: Wed, 6 Nov 2024 15:49:07 -0800 Subject: [PATCH 14/15] Edits --- posts/classical-mechanics/index.qmd | 20 ++++++++++---------- posts/classical-mechanics/styles.css | 22 ++++++++++++++++++++++ 2 files changed, 32 insertions(+), 10 deletions(-) create mode 100644 posts/classical-mechanics/styles.css diff --git a/posts/classical-mechanics/index.qmd b/posts/classical-mechanics/index.qmd index 7ccc5d2..c3d1474 100644 --- a/posts/classical-mechanics/index.qmd +++ b/posts/classical-mechanics/index.qmd @@ -2,8 +2,9 @@ title: "Pythagorean Time Dilation" subtitle: "A Summary of Classical Mechanics" author: John Vivian -date: 2024-10-15 +date: 2024-11-06 categories: [Math, Physics] +image: "imgs/krusty.png" format: html: css: styles.css @@ -50,7 +51,10 @@ This overview is roughly structured by the concepts I found most profound while I’m neither a mathematician or physicist but do my best to clarify the meaning of each variable and operator in the equations introduced. ::: -# Absolute Power, Absolute Acceleration +# Absolute Acceleration +> _Speed has never killed anyone. Suddenly becoming stationary, that's what gets you_ [^9] + +[^9]: Jeremy Clarkson Imagine yourself floating in deep space, with only distant stars as a backdrop. Would you be able to tell if you were moving? Surprisingly, no. As Einstein demonstrated, **velocity only has meaning relative to another reference frame**. If two people drift past each other, neither can determine who is “stationary” or “moving” in any absolute sense—both could be moving, or one could be stationary. In empty space, without other points of reference, velocity becomes entirely relative. @@ -58,9 +62,6 @@ Oddly, _acceleration is different_. If you were suddenly accelerated in your spa In fact, you can create your own “accelerometer” with something as simple as a spring and weight. Hang the weight from a spring, and if you accelerate left or right, the weight will stretch the spring in the opposite direction. This setup works because acceleration produces a noticeable force, while constant velocity does not. - - - ```{python} #| label: fig-spring_weight_visualization-dffad #| fig-cap: "Spring with a hanging weight, stretching left as the system accelerates to the right." @@ -234,7 +235,7 @@ $$v^2 = v_0^2 + 2a(x-x0)$$ ::: ## The Universal Law of Gravitation -> Quote here +> Not the Deepak Chopra variety Newton’s Law of Gravitation describes the gravitational force between two masses, expressing how every mass exerts an attractive force on every other mass in the universe: @@ -318,7 +319,7 @@ $$F = G\frac{m_1m_2}{r^2}$$ Why is $r$ squared? Why not quadrupled or halved? The inverse square relationship emerges as a property of living in three dimensions. The volume of a sphere increases with the square of the radius: $4\pi r^2$. # Maxwell's Equations -> Quote here +> The Original Field Science Maxwell's Equations elegantly describe the behavior of electric and magnetic fields, revealing that these two forces are not separate but deeply interconnected. They provide a unified framework for understanding how electric charges and currents create fields that interact, propagate, and influence each other, ultimately giving rise to light itself. @@ -900,12 +901,12 @@ An unexpected consequence of the **Michelson-Morley Experiment** of 1887---while ### The Speed of Light, $c$, and the Foundations of Special Relativity -Einstein realized that, to be consistent with Maxwell’s equations for electromagnetism, light’s speed must remain constant across all frames of reference. This insight forced a rethinking of space and time. The concepts of absolute time (a universal “clock”) and absolute space (a universal grid) had to be discarded. Instead, space and time are intertwined in such a way that the speed of light remains constant—a revolutionary shift that led to the notion of spacetime. +Einstein realized that, to be consistent with Maxwell’s equations for electromagnetism, light’s speed must remain constant across all frames of reference. This insight forced a rethinking of space and time. The concepts of absolute time (a universal “clock”) and absolute space (a universal grid) had to be discarded. Instead, space and time are intertwined in such a way that the speed of light remains constant—a revolutionary shift that led to the notion of **spacetime**. 1. Invariance of $c$: The speed of light in a vacuum is the same for all observers, regardless of their motion. This means that whether you’re standing still, driving a car, or hurtling through space, $c$ is always the same. 2. Space and Time Adapt: For this invariance to hold, space and time themselves must adapt to keep the speed of light constant. If you move faster, your time slows down and lengths contract relative to slower-moving observers. 3. Relativity of Simultaneity: Events that appear simultaneous to one observer may not be simultaneous to another moving observer. This idea challenges the intuitive notion of a shared “now” and suggests that time flows differently depending on motion. An event in one observer's past can be in another's future depending on how fast they are moving relative to one another. -4. More directly, we now know this also limits _informations_ ability to propagate faster than speed of light. Quantum entanglement, counter-intuitively, does _not_ violate this principle. +4. More directly, we now know this also limits _informations_ ability to propagate faster than $c$. Quantum entanglement, counter-intuitively, does _not_ violate this principle. ### Time is Just Another Coordinate @@ -1373,7 +1374,6 @@ Kruskal coordinates provide a profound visualization of how **general relativity - **Causality** inside the event horizon is altered: what we consider “future” becomes tied to an inevitable destination (the singularity). # Timeline of The Universe -> The End $Ga$ is **Giga-Annum** or 1 billion Earth years diff --git a/posts/classical-mechanics/styles.css b/posts/classical-mechanics/styles.css new file mode 100644 index 0000000..fe67a97 --- /dev/null +++ b/posts/classical-mechanics/styles.css @@ -0,0 +1,22 @@ +/* Increase font size, node padding, and spacing for Mermaid timeline */ +.mermaid svg { + font-size: 18px; /* Increases font size */ +} + +.mermaid .label text { + font-size: 16px; /* Increases node label font size */ +} + +.mermaid .node rect, +.mermaid .node circle, +.mermaid .node ellipse { + padding: 8px; /* Adds padding inside nodes */ +} + +.mermaid .edgeLabel text { + font-size: 14px; /* Increases font size for edge labels */ +} + +.mermaid .label rect { + width: 200px; /* Adjusts width of nodes */ +} From 4c6b8340f64740b5cbcacc9e6d2d6d89c7aeb531 Mon Sep 17 00:00:00 2001 From: jvivian Date: Wed, 6 Nov 2024 15:49:34 -0800 Subject: [PATCH 15/15] Docs render --- docs/about.html | 300 ++- docs/index.html | 4 +- docs/posts/classical-mechanics/index.html | 46 +- docs/search.json | 12 +- ...p-44266c55e4ed57b741d21de50a2a288a.min.css | 12 + .../bootstrap/bootstrap-dark.min.css | 12 +- docs/site_libs/bootstrap/bootstrap-icons.css | 148 +- docs/site_libs/bootstrap/bootstrap-icons.woff | Bin 164168 -> 176200 bytes docs/site_libs/bootstrap/bootstrap.min.js | 6 +- docs/site_libs/quarto-diagram/mermaid-init.js | 35 +- docs/site_libs/quarto-diagram/mermaid.min.js | 2189 ++++++++++++++++- docs/site_libs/quarto-html/anchor.min.js | 6 +- docs/site_libs/quarto-html/popper.min.js | 4 +- ...hting-2486e1f0a3ee9ee1fc393803a1361cdb.css | 205 ++ ...-dark-6561bbde787c299e21493071c8edc6ff.css | 189 ++ docs/site_libs/quarto-html/quarto.js | 49 +- docs/site_libs/quarto-listing/list.min.js | 2 +- .../quarto-listing/quarto-listing.js | 7 +- docs/site_libs/quarto-nav/quarto-nav.js | 58 +- .../quarto-search/autocomplete.umd.js | 4 +- docs/site_libs/quarto-search/quarto-search.js | 230 +- 21 files changed, 3305 insertions(+), 213 deletions(-) create mode 100644 docs/site_libs/bootstrap/bootstrap-44266c55e4ed57b741d21de50a2a288a.min.css create mode 100644 docs/site_libs/quarto-html/quarto-syntax-highlighting-2486e1f0a3ee9ee1fc393803a1361cdb.css create mode 100644 docs/site_libs/quarto-html/quarto-syntax-highlighting-dark-6561bbde787c299e21493071c8edc6ff.css diff --git a/docs/about.html b/docs/about.html index 55e151d..4ca83ea 100644 --- a/docs/about.html +++ b/docs/about.html @@ -2,12 +2,12 @@ - + -blog - About +About – blog

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