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%%%%%%%%%%%%%%%%% CLASSICAL MECHANICS %%%%%%%%%%%%%%%%
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\begin{center}
{\bf ADVANCED DYNAMICS --- PHY--4241/5227}\\\vspace {2mm} 
{\bf HOMEWORK 3}\\%
\vspace {3mm} (\today) \\
\vspace {1mm}{Due on Monday, January 27, 2003} \\ 
\end{center}
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\centerline{\bf PROBLEM 7}
\centerline{\small{(Example 7-12 Marion and Thornton)}}
\medskip
Consider a spherical ({\it i.e.,} three-dimensional)
pendulum of mass $m$ and length $b$, as displayed in
Figure 7-10 of Marion and Thornton.
\begin{description}
\item{a)} Using spherical coordinates, obtain the Lagrangian
          of the system $L(\theta,\phi,\dot{\theta},\dot{\phi})$.
\item{b)} Using the Legendre transformation
          $$
            H(\theta,\phi,p_{\theta},p_{\phi})=
	    \dot{\theta}p_{\theta}+\dot{\phi}p_{\phi} -
	    L(\theta,\phi,\dot{\theta},\dot{\phi})\;,
	  $$  
	  construct the Hamiltonian of the system and show 
	  that it is identical to the energy $E\!=\!T\!+\!V$.
\item{c)} Using the method of the effective potential, obtain
          the conditions for which the pendulum will move in 
	  an orbit of constant height, that is, of constant
	  $\theta\!=\!\theta_{0}$.
\end{description}
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\centerline{\bf PROBLEM 8}
\centerline{\small{(Problem 7-30 Marion and Thornton)}}
\medskip
Consider any two continuous functions of the generalized
coordinates and momenta (and possible time) $g(q_{k},p_{k},t)$ 
and $h(q_{k},p_{k},t)$. The {\bf Poisson bracket} of $g$ and 
$h$ is defined by
$$
  [g,h]\equiv\sum_{k} 
    \left(\frac{\partial g}{\partial q_{k}}
          \frac{\partial h}{\partial p_{k}} -
	  \frac{\partial g}{\partial p_{k}}
	  \frac{\partial h}{\partial q_{k}}\right) \;.
$$
Verify the following properties of the Poisson brackets:
\begin{description}
\item{a)} $\displaystyle{\frac{dg}{dt}}=[g,H]+
           \displaystyle{\frac{\partial g}{\partial t}}\;.$
\item{b)} $\dot{q}_{j}=[q_{j},H]\;; \quad \dot{p}_{j}=[p_{j},H]\;.$
\item{c)} $[q_{i},q_{j}]=[p_{i},p_{j}]=0\;; \quad 
           [q_{i},p_{j}]=\delta_{ij}\;.$
\end{description}
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\centerline{\bf PROBLEM 9}
\medskip
\begin{description}
\item{a)} Show that if $f$, $g$, and $h$ are arbitrary
          functions of $q$ and $p$ (and possible time) 
	  then the following relations among Poisson 
	  brackets hold true:
          \begin{eqnarray}
           [fg,h] &=& [f,h]g + f[g,h] \;, \cr
           [f,gh] &=& [f,g]h + g[f,h] \;. \nonumber
          \end{eqnarray}

\item{b)} Show that the Poisson bracket between an
          {\bf arbitrary} function of
          $r\!=\!\sqrt{(x^2+y^2+z^2)}$ and the
          momentum ${\bf p}$ is given by
          $$
            [f(r),p_{i}]=\frac{x_{i}}{r}f'(r) \;.
          $$

\item{c)} Assuming the validity of the fundamental
          Poisson-bracket relations between coordinates
          and momenta
          $$
           [x_{i},x_{j}]=[p_{i},p_{j}]=0\;; \quad
           [x_{i},p_{j}]=\delta_{ij} \;,
          $$
          compute the following Poisson brackets:
          $$
           [x_{i},L_{j}]\;,\; [p_{i},L_{j}] \;,\;
           {\rm and}\;\;      [L_{i},L_{j}] \;,
          $$
          where ${\bf L}={\bf r}\times{\bf p}$ is
          the angular momentum of the system.
\end{description}
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