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\begin{center}
{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 6}\\%
\vspace {3mm} (February 11, 2009) \\
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\begin{description}
\item{(24)} 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:
\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}
Due February~13 before class (10 points). % Read M\&T chapter 7.12 (due 

\item{(25)} Consider two point particles with a Lagrangian (due February~13
            in class)
$$ {\cal L} = \frac{m_1}{2}\vec{v}_1^{\,2}+\frac{m_2}{2}\vec{v}_2^{\,2}
           - U(r)\,,~~~r=|\vec{r}|\,,\ \vec{r}=\vec{r}_1-\vec{r}\ . $$
\begin{enumerate}
  \item Is the energy of this system conserved (with reason) (2 points)?
  \item Define the center of mass vector $\vec{R}=\vec{R}(t)$ by 
        $M\dot{\vec{R}}=m_1\dot{\vec{r}}_1+m_2\dot{\vec{r}}_2$, where 
        $M=m_1+m_2$ is the total mass, and find the general solution 
        for $\vec{R}(t)$ (2~points).
  \item Use the special solution $\vec{R}=(m_1\vec{r}_1+m_2\vec{r}_2)/M$, 
        to express $\vec{r}_1$ and $\vec{r_2}$ through $\vec{R}$ and
        $\vec{r}$ (3 points). \medskip

The center of mass (cm) systems is defined by $\vec{R}(t)=0$. Show 
the following equalities in the system . \medskip

  \item $T_{\rm cm} = m_1\,\vec{v}_1^{\,2}/2+m_2\,\vec{v}_2^{\,2}/2 
  = \mu\,\vec{v}^{\,2}/2$ with $\vec{v}=\dot{\vec{r}}$ and $\mu$ the 
  reduced mass. Express $\mu$ through $m_1$, $m_2$ and $M$ (3 points).
  \item $\vec{L}=\vec{r}_1\times\vec{p}_1+\vec{r}_2\times\vec{p}_2=
         \mu\,\vec{r}\times\vec{v}$ (2 points).
\end{enumerate}

\end{description}

% \begin{figure}[tb] \vspace{5pc}
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% \caption{Ladder for assignment~14.} \label{fig_lader} \end{figure}
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