%This is a template for LaTeXing homework for
%the Winter 2008 classical mechanics course by John Baez
%Please fill in the number and date of the lecture the homework
%came from in the slot below, and say who you are!
%Please fill in the number and date of the lecture in the appropriate
%slot below, and say who is taking these notes.
%Also: use the macros below for common physics and math symbols!
%
%In particular: use \R for the real numbers!
%
%Use \maps for the colon in the notation for functions:
%instead of f: X \rightarrow Y, please write f \maps X \to Y.
%
%Use \define for terms being defined,
%as in: ``We define the \define{position} to be...''
\documentclass{article}
\usepackage{amsfonts,amssymb}
%\usepackage{latexsym}
\hfuzz=6pt
% common physics symbols - use these macros!
\newcommand{\q}{q} %position
\newcommand{\p}{p} %momentum
\newcommand{\E}{E} %energy
\newcommand{\T}{T} %kinetic energy
\newcommand{\V}{V} %potential energy
\newcommand{\J}{J} %angular momentum
% common math symbols - use these macros
\newcommand{\maps}{\colon} %correct symbol for colon in f: X -> Y
%write this as: f \maps X \to Y
\newcommand{\R}{{\mathbb R}} %real numbers
\newcommand{\C}{{\mathbb C}} %complex numbers
\newcommand{\Z}{{\mathbb Z}} %integers
\renewcommand{\O}{{\rm O}} %orthogonal group
\newcommand{\SO}{{\rm SO}} %special orthogonal group
\newcommand{\so}{{\frak{so}}} %special orthogonal Lie algebra
% use \define for defined terms:
\newcommand{\define}[1]{{\bf #1}}
\newtheorem{thm}{Theorem}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{rem}[thm]{Remark}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{defn}[thm]{Definition}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\ba}{\begin{eqnarray}}
\newcommand{\ea}{\end{eqnarray}}
\newcommand{\ban}{\begin{eqnarray*}}
\newcommand{\ean}{\end{eqnarray*}}
\newcommand{\barr}{\begin{array}}
\newcommand{\earr}{\end{array}}
\textwidth 6in
\textheight 8.5in \evensidemargin .25in
\oddsidemargin .25in
\topmargin .25in
\headsep 0in
\headheight 0in
\footskip .5in
\pagestyle{plain}
\pagenumbering{arabic}
\begin{document}
\begin{center}
{\large Classical Mechanics Homework\\}
{\small February 21, 2008 \\}
{\small John Baez}
\end{center}
\section*{Angular Momentum and Rotations}
In this problem we will see that angular momentum generates
rotations for a particle in $\R^n$. We begin by recalling
a bit about rotations. Let $\O(n)$ be the {\bf orthogonal group}: the
group of all linear transformations of $\R^n$ that preserve
distances. We can describe an element $R \in \O(n)$ as a real
$n \times n$ matrix that is {\bf orthogonal}, meaning
\[ OO^* = O^* O = I \]
where $O^*$ is the adjoint of the matrix $O$ and $I$ is the
identity matrix.
\vskip 1em \noindent
We can define the {\bf exponential} of any $n \times n$ real
matrix $A$ to be the matrix defined by
\[ \exp(A) = \sum_{n = 0}^\infty \frac{A^n}{n!} \]
(This series always converges.)
Some easy calculations show that
\[ \exp((s+t)A) = \exp(sA) \exp(tA) \]
for all $s,t \in \R$. Also, the entries of the matrix $\exp(tA)$
are smooth functions of $t \in \R$.
\vskip 1em \noindent
1. Suppose that $A$ is {\bf skew-adjoint}, meaning $A^* = -A$.
Show that $\exp(t A) \in \O(n)$ for all $t \in \R$.
\vskip 1em \noindent
The group $\O(n)$ includes both rotations and reflections.
In particular, $\O(n)$ consists of two connected components --- the
component where $\det(R) = 1$ and the component where $\det(R) = -1$.
We define the {\bf rotation group} or {\bf special orthogonal
group} $\SO(n)$ to be the subgroup consisting of all $R \in \O(n)$
with $\det(R) = 1$. This subgroup only includes rotations.
A continuous curve can never go from one component to another.
So, if $A$ is skew-adjoint, $\exp(tA)$ must actually lie in
$\SO(n)$ for all $t$.
\vskip 1em \noindent
We define $\so(n)$ to be the set of all skew-adjoint real
$n \times n$ matrices. This set $\so(n)$ is actually a
Lie algebra, since it is a vector space closed under the
bracket operation $[x,y] = xy -yx$. It is called the
{\bf Lie algebra of the rotation group}.
\vskip 1em \noindent
Now, let $\R^{2n}$
be the phase space for a particle in $\R^n$. A point
$(\q,\p) \in \R^{2n}$ describes the particle's {\bf position}
$\q \in \R^n$ and {\bf momentum} $\p \in \R^n$. The algebra
of smooth real-valued functions $C^\infty(\R^{2n})$ becomes a
Poisson algebra with
\[ \{ F, G \} = \sum_{i = 1}^n
\frac{\partial F}{\partial p_i} \frac{\partial G}{\partial q_i} -
\frac{\partial G}{\partial p_i} \frac{\partial F}{\partial q_i}
.\]
\vskip 1em \noindent
2. Given $A \in \so(n)$, let
\[ \phi \maps \R \times \R^{2n} \to \R^{2n} \]
be given by
\[ \phi(t,q,p) = (\exp(tA)q, \exp(tA)p) .\]
Using the facts I've told you, show that $\phi$ is a flow.
\vskip 1em \noindent
{\it (For example, in 3 dimensions, this flow would rotate both the
position and the momentum about some axis.)}
\vskip 1em \noindent
3. Given $A \in \so(n)$,
define an observable $F \in C^\infty(\R^{2n}$ by
\[ F(q,p) = \sum_{i,j = 1}^n A_{ij} (q_i p_j - q_j p_i) .\]
Show that some multiple of $F$ generates the flow $\phi$ defined
above.
\vskip 1em \noindent
{\it (I say `some multiple' because you may need a factor
of $\frac{1}{2}$ or a minus sign or something in front of $F$
to make this calculation work. I leave that to you!)}
\vskip 1em \noindent
{\bf The moral:} The observable that generates the flow $\phi$ is called
{\bf angular momentum in the $A$ direction}. But beware:
$A$ is not a vector in $\R^n$! It's a matrix in $\so(n)$!
For $n = 3$ we have an isomorphism
\[ \so(n) \cong \R^n \]
so we can talk about angular momentum in some direction
$v \in \R^n$. But, this is not true in any other dimension
(except $n = 0$)!
\vskip 1em \noindent
4. When $n = 3$, the observable
\[ F(q,p) = q_1 p_2 - q_2 p_1 \]
is usually called {\bf angular momentum in the $z$ direction}
and denoted $J_z$. What flow does this observable generate?
\end{document}