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\begin{center}
{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 5}\\%
\vspace {3mm} (February 1, 2010) \\
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\noindent Read M\&T chapter 7.10 (due February~3).

\begin{description}

\item{(20)} Liouville's Theorem: We define the velocity in phase space
as a $2n$-dimensional vector $\vec{v}=(\dot{q}_1,\dots,\dot{q}_n,
\dot{p}_1,\dots,\dot{p}_n)$. A large collection of particles can be 
described by their density in phase space $\rho(q_1,\dots,q_n,p_1,
\dots,p_n)$. If their are no sources or sinks, we have a conserved
current
\begin{equation} \label{cc}
  \frac{\partial\rho}{\partial t}+\nabla\left(\vec{v}\,\rho\right)=0
\end{equation} 
$$ {\rm where}\ \nabla=\left(\frac{\partial\ }{\partial q_1},\dots,
  \frac{\partial\ }{\partial q_n},\frac{\partial\ }{\partial p_1},
  \dots, \frac{\partial\ }{\partial p_n}\right)\ {\rm is\ the\ gradient
  \ in\ phase\ space.}  $$
(a) Expand (\ref{cc}) in sums of partial derivatives (you get five
terms when you keep coordinates and momenta in separate contributions).
(b) Use Hamilton's equations of motion to show that two terms cancel 
out. (c) Combine the remaining terms to
\begin{equation} 
   \frac{d\rho}{dt} = 0~~{\rm (Liouville's\ Theorem).} 
\end{equation} 
Due February~5 in class (6 points). Read M\&T chapter 7.12.

\item{(21)} Poisson brackets are defined by
$$[g,h] = \sum_k \left( 
  \frac{\partial g}{\partial q_k}\,\frac{\partial h}{\partial p_k}- 
  \frac{\partial h}{\partial q_k}\,\frac{\partial g}{\partial p_k}
  \right) $$
where $g$ and $h$ are functions of $q_i$, $p_i$ and, possibly, $t$.
Show the following properties (due February~10 before class, 10 points): 
\begin{enumerate}
\item $$ \frac{dg}{dt} = [g,H]+\frac{\partial g}{\partial t}\ .$$
\item $$ \dot{q}_j = [q_j,H]\ .$$
\item $$ \dot{p}_j = [p_j,H]\ .$$
\item $$ [x_{i},x_{j}]=[p_{i},p_{j}]=0\;; \quad
         [x_{i},p_{j}]=\delta_{ij} \;, $$
\item $$ [x_{i},L_{j}] = \epsilon_{ijk}x_k\;,\; 
         [p_{i},L_{j}] = \epsilon_{ijk}p_k\;,\; {\rm and}\;\;  
         [L_{i},L_{j}] = \epsilon_{ijk}L_k\;,\; \;, $$
          where the Einstein summation convention is used and 
          $L_j=\epsilon_{jkl}\,x_k\,p_l$ is the $i$th component
          of the angular momentum of the system.
\end{enumerate}

\end{description}

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