# Theoretical Minimum Practice Problems

The Theoretical Minimum is a great book for advanced classical mechanics. It contains pretty much everything you need to know for university mechanics, but it doesn’t really supply enough practice problems. For pretty much everything in the book, the results are much less important than the methods for obtaining the results. If somehow you memorize the result of Noether’s theorem as “If […], then $\frac{d}{dt}\sum_i (\cdots)=0$”, not only will you have wasted time memorizing results, you’ll be totally clueless when you have to do similar things in quantum mechanics. You really do have to know this stuff inside and out!

Here are a list of practice problems you should be able to complete without referring to the book. Once you can sit in a library and go through this list from start to finish, you’ll have got enough out of the book.

## Lecture 6 problems

- Derive the Euler-Lagrange equations from the principle of least action.
- Write the Lagrangian of a free particle using coordinates in a rotating reference frame.
- Use the previous Lagrangian to derive fictitious corioils/centrifugal forces.
- Write $L=\frac{m}{2}(\dot{x}_ 1^2+\dot{x} _2^2)-V(x _1-x _2)$ and write out the change of variables to $x _ +=\frac{x _ 1+x _ 2}{2}$ and $x _ -=\frac{x _ 1-x _ 2}{2}$.

## Lecture 7 problems

- Show that if $L=\frac{m}{2}(\dot{x}_1^2+\dot{x}_2^2)-V(x_1^2+x_2^2)$, the infinitesimal translations $x \mapsto x+y \delta$, $y\mapsto y-x\delta$ leave $L$ unchanged (to first order in $L$).
- Consider $L({\bf q},\dot{\bf q})$ and suppose each $q_i(t)$ satisfies the EL equations. Recall $p_i=\frac{\partial L}{\partial \dot{q}_i}$. Make an infinitesimal transformation $q_i\mapsto q_i+f_i(q)\delta$, and show that if $L$ is unchanged to first order, then the quantity $Q=\sum_i p_i f_i(q)$ is conserved over time.
- Work out the Lagrangian for a double pendulum in a gravitational field. Find the conjugate momenta, EL equations, and rate of change of angular momentum.

## Lecture 8 problems

- Use $H=\sum_i (p_i \dot{q}_i)-L$ to show $\dot{H}=-\frac{\partial L}{\partial t}$.
- Given the harmonic oscillator $L=\frac{1}{2}m \dot{x}^2-\frac{1}{2}kx^2$, substitute $q=(km)^{\frac{1}{4}}x$, $\omega=\sqrt{\frac{k}{m}}$, simplify the Lagrangian, then find the Hamiltonian, and then find Hamilton’s equations.
- Derive Hamilton’s equations by expanding $\delta H$ two different ways.

## Lecture 9 problems

- Show that if ${\bf v}=(\dot{q}_1,\dot{q}_2,\cdots,\dot{p}_1,\dot{p}_2\cdots)$, where the $q,p$ satisfy Hamilton’s equations, then $\nabla \cdot {\bf v}=0$.
- Use the Poisson bracket axioms $\{A,C\}=-\{C,A\}$, $\{A,A\}=0$, $\{k A,C\}= k\{A,B\}$, $\{A+B,C\}=\{A,C\}+\{B,C\}$, $\{AB,C\}=B\{A,C\}+A\{B,C\}$, and $\{q_i,p_j\}=\delta_{ij}$, as well as the theorems $\{F,p_i\}=\frac{\partial F}{\partial q_i}$ and $\{F,q_i\}=-\frac{\partial F}{\partial p_i}$, prove that $\dot{q}=\{q,H\}$ and $\dot{p}=\{p,H\}$ for a particle-in-potential Hamiltonian is equivalent to Newton’s laws of motion.

## Lecture 10 problems

- Given $L_k=\varepsilon_{i j k} q_i p_j$, find $\{q_i,L_3\}$ through both the Poisson bracket axioms, and through the derivative definition. (the summation convention is taken on that first equation)
- Show $\{x_i,L_j\}=\varepsilon_{i j k}x_k$.
- Show that if $H=\omega L_3$ then $\dot{L}_3=0$, $\dot{L}_1=-\omega L_2$, and $\dot{L}_2=\omega L_1$.