Advanced Classical Physics

Learn some of the most fascinating classical physics Newton didn’t know, including chaos, relativity, waves, and field theories.

Teacher: Robert Jones

Description

Classical physics has grown to encompass many topics since its early development. We begin by studying how systems with seemingly few "moving parts," such as a simple pendulum, can exhibit surprisingly complicated behaviors such chaos and hysteresis. Then we'll examine systems with many degrees of freedom, including magnets and waves, to see how emergent behavior arises from their parts. Lastly, turning up the degrees of freedom to infinity, we will build field theories, such as those that underlie electromagnetism and gravity.

Some strange behaviors, such as topological defects, are illustrated in even more esoteric forces with applications in cosmology. Guided by our study of these systems, we will learn about the Lagrangian formulation of mechanics and both special and general relativity, and we'll explore the importance of deep notions such as symmetry, nonlinearity, and degrees of freedom.

Tentative list of topics covered

Zero degrees of freedom (discrete systems): time evolution and Liouville's theorem, cellular automata, entropy and information, the Ising model.
A few degrees of freedom: linear time invariant systems, chaos, linearization and perturbation theory, Lagrangian and Hamiltonian mechanics, special relativity.
Many degrees of freedom: Fluids and waves, symmetry, classical field theories with applications in general relativity and cosmology, gauge theories

For the application...

Prerequisites

Physics and calculus, both at the honors, AP, or IB level; or equivalent knowledge.

Relevant experience

Please list your coursework and grades in any advanced mathematics and physics classes you have taken. Also include any extracurricular experiences (including things you did informally on your own!) you have had related to math, physics, or computer science.

Core-specific application question

You may also download these questions as a PDF.

The application question for this class consists of multiple parts. You should provide responses to all parts. These are designed to require some thought, so don’t be discouraged if it takes you a while to come up with good answers. You can do it!

Do not worry if you cannot answer all of the parts; you are not necessarily under-qualified to take this course if you can't (and probably not over-qualified even if you can).

Please document your thought process clearly. Your work is much more important to us than your final answer.

Feel free to consult references or make use of software packages such as Mathematica, Maple, or Wolfram Alpha, as long as you clearly document which aids you used and where you used them.

Part 1. Mathematics

One of the things we will study in this class is the behavior of systems in which we treat time as a discrete variable. In particular, suppose we pick some initial $$ x_0 $$ with $$ -1 \le x_0 \le 1 $$, and then generate $$ x_1 $$, $$ x_2 $$, and so on according to the rule (called a map)

$$ x_{n+1} = f(x_n), $$ where $$f(x) = 1 - \alpha x^2,$$ and $$\alpha$$ is a parameter between 0 and 2.

(A) A fixed point is a point $$x^*$$ such that $$f(x^*) = x^*$$. For example, when $$\alpha = 0$$, 1 is a fixed point. Find the fixed point $$x^*_\alpha$$ as a function of $$\alpha$$.

(B) A fixed point is stable if, for small $$\varepsilon$$, $$f(x^* + \varepsilon)$$ is closer to $$x^*$$ than $$x^* + \varepsilon$$ is. In other words, points near the fixed point tend toward the fixed point: $$|f(x^* + \varepsilon) - x^*| < |\varepsilon|$$. For which values of $$\alpha$$ is $$x^*_\alpha$$ stable? You may wish to use the fact that $$ f(x+\varepsilon) \approx f(x) + \varepsilon f'(x) $$ for small $$\varepsilon$$.

(C) How does the map behave for other values of $$\alpha$$? Does it vary in a regular way as you increase $$\alpha$$, or does it do something else? Describe the behavior of the map as best as you can. For various values of $$\alpha$$, you may wish to examine the the behavior of the map for large $$n$$.

Part 2. Physics

A particle of mass $$m$$ moves along the $$x$$-axis under the influence of some potential $$V(x)$$. You determine that the time $$t$$ at which the particle passes through a position $$x$$ is given by

$$t(x) = 2 t_0 \ln\left(e^{x/(2x_0)} + \sqrt{e^{x/x_0} - \alpha}\right)$$, where $$\alpha$$ is a constant.

(A) Show that

$$\frac{dt}{dx} = \frac{t_0}{x_0} \frac{e^{x/(2x_0)}}{\sqrt{e^{x/x_0} - \alpha}}$$.

(B) Determine the particle's kinetic energy $$K(x)$$ as a function of $$x$$ and simplify your expression. You may wish to make use of the fact that $$dy/dx = 1/(dx/dy)$$.

Assume that the total energy of the particle is conserved. In other words, the sum of the kinetic energy $$K$$ and potential energy $$V$$ is the total energy $$E$$:

$$E = K(x) + V(x)$$.

(C) Explain why we are free to choose $$E$$ to be anything we want without changing the physics, as long as we make sure to set $$V(x) = E - K(x)$$.

(D) Choose $$E$$ such that for $$x \rightarrow \infty$$, the potential energy goes to zero. What is $$V(x)$$ given this choice of $$E$$?

(E) Determine $$\alpha$$ so that the particle passes through $$x_0$$ when $$t = t_0$$. Given this choice of $$\alpha$$, what are the kinetic energy and velocity of the particle at $$x_0$$?

# Advanced Classical Physics

Learn some of the most fascinating classical physics Newton didn’t know, including chaos, relativity, waves, and field theories.

Teacher: Robert Jones

***

## Description

## Tentative list of topics covered

* **Zero degrees of freedom (discrete systems):** time evolution and Liouville's theorem, cellular automata, entropy and information, the Ising model.
* **A few degrees of freedom:** linear time invariant systems, chaos, linearization and perturbation theory, Lagrangian and Hamiltonian mechanics, special relativity.
* **Many degrees of freedom:** Fluids and waves, symmetry, classical field theories with applications in general relativity and cosmology, gauge theories

***

# For the application...

## Prerequisites

Physics and calculus, both at the honors, AP, or IB level; or equivalent knowledge.

## Relevant experience

## Core-specific application question

_You may also [download these questions as a PDF](https://esp.mit.edu/download/9900e492-3a87-4d20-bab7-3a7c46fdcf3b/phys-appq-part-both.pdf)._

The application question for this class consists of multiple parts. You should provide responses to all parts. These are designed to require some thought, so don’t be discouraged if it takes you a while to come up with good answers. You can do it!

Do not worry if you cannot answer all of the parts; you are not necessarily under-qualified to take this course if you can't (and probably not over-qualified even if you can).

Please document your thought process clearly. Your work is much more important to us than your final answer.

Feel free to consult references or make use of software packages such as Mathematica, Maple, or Wolfram Alpha, as long as you clearly document which aids you used and where you used them.

### Part 1. Mathematics

One of the things we will study in this class is the behavior of systems in which we treat time as a _discrete_ variable. In particular, suppose we pick some initial $$ x_0 $$ with $$ -1 \le x_0 \le 1 $$, and then generate $$ x_1 $$, $$ x_2 $$, and so on according to the rule (called a _map_)
<center>$$ x_{n+1} = f(x_n), $$</center>
where
<center>$$f(x) = 1 - \alpha x^2,$$</center>
and $$\alpha$$ is a parameter between 0 and 2.

(A) A _fixed point_ is a point $$x^\*$$ such that $$f(x^\*) = x^\*$$. For example, when $$\alpha = 0$$, 1 is a fixed point. Find the fixed point $$x^\*_\alpha$$ as a function of $$\alpha$$.

(B) A fixed point is _stable_ if, for small $$\varepsilon$$, $$f(x^\* + \varepsilon)$$ is closer to $$x^\*$$ than $$x^\* + \varepsilon$$ is. In other words, points near the fixed point tend toward the fixed point: $$|f(x^\* + \varepsilon) - x^\*| < |\varepsilon|$$. For which values of $$\alpha$$ is $$x^\*_\alpha$$ stable? You may wish to use the fact that $$ f(x+\varepsilon) \approx f(x) + \varepsilon f'(x) $$ for small $$\varepsilon$$.

(C) How does the map behave for other values of $$\alpha$$? Does it vary in a regular way as you increase $$\alpha$$, or does it do something else? Describe the behavior of the map as best as you can. For various values of $$\alpha$$, you may wish to examine the the behavior of the map for large $$n$$.

### Part 2. Physics

A particle of mass $$m$$ moves along the $$x$$-axis under the influence of some potential $$V(x)$$. You determine that the time $$t$$ at which the particle passes through a position $$x$$ is given by
<center>$$t(x) = 2 t_0 \ln\left(e^{x/(2x_0)} + \sqrt{e^{x/x_0} - \alpha}\right)$$,</center>
where $$\alpha$$ is a constant.

(A) Show that
<center>$$\frac{dt}{dx} = \frac{t_0}{x_0} \frac{e^{x/(2x_0)}}{\sqrt{e^{x/x_0} - \alpha}}$$.</center>

(B) Determine the particle's kinetic energy $$K(x)$$ as a function of $$x$$ and simplify your expression. You may wish to make use of the fact that $$dy/dx = 1/(dx/dy)$$.

Assume that the total energy of the particle is conserved. In other words, the sum of the kinetic energy $$K$$ and potential energy $$V$$ is the total energy $$E$$:
<center>$$E = K(x) + V(x)$$.</center>

(C) Explain why we are free to choose $$E$$ to be anything we want without changing the physics, as long as we make sure to set $$V(x) = E - K(x)$$.

(D) Choose $$E$$ such that for $$x \rightarrow \infty$$, the potential energy goes to zero. What is $$V(x)$$ given this choice of $$E$$?

(E) Determine $$\alpha$$ so that the particle passes through $$x_0$$ when $$t = t_0$$. Given this choice of $$\alpha$$, what are the kinetic energy and velocity of the particle at $$x_0$$?

Last modified by lchiou on March 25, 2014 at 01:29 a.m.