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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 4}\\%
\vspace {3mm} (January 26, 2011) \\
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Read Landau-Lifshitz \textsection 10 and \textsection 11. Due January 28.

\begin{description}

\item{(14a)} Use the Einstein summation convention in the following.
Show that
$$ L_i = \epsilon_{ijk}\,x_j\,p_k = [\vec{r}\times\vec{p}\,]_i~~
   {\rm holds\ for}~~i=1,2,3.$$
Due in class (2 points). 

\item{(14b)} Express $\epsilon_{12k}\,\epsilon_{lmk}$ in terms 
of Kronecker delta and then $\epsilon_{ijk}\,\epsilon_{lmk}$.
Due in class (2 points). 
\bigskip

\item{(14c)} 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~2 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 $j$th component
          of the angular momentum of the system.
\end{enumerate}

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Landau-Lifshitz: Skip \textsection 12, read \textsection 13 (due 
February 2) and \textsection 14 (due February 4). 

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