\documentclass[a4paper,12pt]{article}
\title{Mechanics Questions}
\date{\today}
\author{Mark Allen}
\usepackage{amssymb}
\newcommand{\C}{\mathbb{C}}
\newcommand{\KE}{\frac{m}{2}\dot{\bar{r}}^2} 
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}

\newenvironment{proof}[1][Proof]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
\newenvironment{definition}[1][Definition]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
\newenvironment{example}[1][Example]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
\newenvironment{remark}[1][Remark]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}

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\begin{document}


\maketitle

\section{Introduction}
This is an attempt to create a list of all possible ``give definition'' or ``prove theorem'' type questions that can be asked based on the notes for the course so far. It includes all exercise/problem type questions given so far in lectures, but doesn't include any homework exercises (yet). The emphasis is on providing a complete overview of what has been covered in the course so far, in chronological order, as well as an easy way to test how well the material is understood, and so I make no attempt to mimic realistic `exam' questions, nor do I attempt to divide the questions into easy and hard. For the type of questions you'll actually see on an exam, see one of the several other lists of questions available.

I intend to make one of these for at least each of the Maths courses, and possibly for the Physics classes too.
\subsection{Week 1}
This week introduced foundational concepts like generalised coordinates, Einstein notation and the degrees of freedom of a system. We were also introduced to the principle of least action, and the week culminated in a derivation of the Euler-Lagrange equations.
\begin{enumerate}
\item
Explain how the coordinates of a point are given in Einstein notation.
\item
Find the metric of $\mathbb{R}^3$ in cartesian, cylindrical and polar coordinates.
\item
Define the degrees of freedom of a system.
\item
How is the number of degrees of freedom of a system determined?
\item
What is the Lagrangian of a system?
\item
What is the action of a system?
\item
State Hamilton's principle of least action.
\item
Derive (from first principles) the equations of motion for the Lagrangian \[ L = \KE \]
\item
Derive (from first principles) the equations of motion for the Lagrangian \[ L = \frac{m}{2}\ddot{\bar{r}}^2 + \bar{F}(t)\dot{\bar{r}}\]
\item
Suppose we have some function $U = U(\bar{r}_1, \bar{r}_2, \dots , \bar{r}_n)$. What is $\delta U$? (i.e. Show you can do the partial derivative properly.)
\item
Derive the Euler-Lagrange equations.
\end{enumerate}

\subsection{Week 2}
We looked at some stuff relating to the interactions of systems. We also looked at the general method for finding the Lagrangian of a system with constrained motion. The week culminated in the statement of Noether's Theorem.  I'll probably completely rewrite this section once I actually understand what's going on.
\begin{enumerate}
\item
Suppose we have two systems A,B. How do we add together the Lagrangians for the two systems?
\item
What can we say about the multiplicative properties of the Lagrangian?
\item
What can we say about the Lagrangian $\tilde{L}(q,\dot{q},t) = L(q,\dot{q},t) + \frac{d}{dt}f(q,t)$ and why?
\item
Show how this allows us to add any constant to potential energy without altering the EOMs.
\item
For the Lagrangian \[L = \sum_{a=1}^{N}\frac{m_a}{2}\bar{v}^2_a - U(\bar{r}_1,\dots,\bar{r}_n)\]
\begin{enumerate}
\item
What are some of the implications of the fact that U has no explicit time dependence?
\item
Derive the EOM.
\item
Rewrite the KE term in terms of arbitrary generalised coordinates.
\end{enumerate}
\item
What sort of form would the Lagrangian for two interacting systems A and B have?
\item
What can we say about the motion of B if the reaction of B on the action of A is negligible?
\item
What is the method for finding the Lagrangian of a system with constrained motion?
\item
Find the Lagrangian for a coplanar double pendulum in a uniform gravitational field.
\item
What is an Integral of Motion?
\item
How many Integrals of Motion are there?
\item
What is an additive Integral of Motion and why is it useful?
\item
In rough terms, what does Noether's Theorem state?

\end{enumerate}

\subsection{Week 3}
This week we continued looking at the symmetries and associated conservation laws of systems, as well as proving Noether's theorem.
\begin{enumerate}
\item
Something something show linear momentum / conservation \emph{FIX THIS}
\item
Prove Noether's Theorem


\end{enumerate}
\subsection{Week 4}
TODO
\subsection{Week 5}
TODO
\subsection{Week 6}

\subsection{Reading Week}
TODO
\end{document}
