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%%%%%%%%%%%%%%%%% CLASSICAL MECHANICS %%%%%%%%%%%%%%%%
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{\bf ADVANCED DYNAMICS --- PHY--4241/5227}\\\vspace {2mm} 
{\bf HOMEWORK 4}\\%
\vspace {3mm} (\today) \\
\vspace {1mm}{Due on Monday, February 3, 2003} \\ 
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\centerline{\bf PROBLEM 10}
\medskip
A particle of mass $m$ and electric charge $q$ moves under the 
influence of a constant magnetic field of the form
${\bf B}({\bf r})\!=\!B_0\hat{\bf z}$. Obtain the most general 
solution for the velocity ${\bf v}(t)$ using Newton's second
law of motion in combination with the Lorentz force. That is,
$$
   {\bf F}=m{\dot{\bf v}}=\frac{q}{c}{\bf v}\times{\bf B}\;.
$$
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\bigskip
\centerline{\bf PROBLEM 11 and 12}
\medskip
The Lagrangian for a particle of mass $m$ and electric charge 
$q$ moving under the influence of a magnetic (but no electric) 
field is given by
$$
   L({\bf r},\dot{\bf r})=\frac{1}{2}m\dot{\bf r}^{2}
                         +\frac{q}{c}\dot{\bf r}\cdot
                          {\bf A}({\bf r})\;,
$$
where ${\bf A}({\bf r})$ is the vector potential. Assume that 
the magnetic field is constant and given by 
${\bf B}({\bf r})\!=\!B_0\hat{\bf z}$. 
\begin{description}
\item{a)} Show that for such a constant magnetic field 
          the vector potential can be written in the form
          ${\bf A}({\bf r})=\frac{1}{2}
             ({\bf B}\times{\bf r})\;.$ That is, show that
	  such a vector potential satisfies: 
	  $\nabla\!\times\!{\bf A}\!=\!{\bf B}\;.$

\item{b)} Construct the Hamiltonian of the system in terms of the
          Cartesian coordinates and the corresponding canonical 
	  momenta of the particle.

\item{c)}  Denoting by 
           ${\pmbf\pi}\!\equiv\!m\dot{\bf r}\!=\!m{\bf v}$
           the ``mechanical'' momentum of the particle,
           evaluate the following Poisson brackets:
             $$
              [\pi_x,\pi_y]\;,\; [\pi_y,\pi_z]\;,\;
              [\pi_z,\pi_x]\;.
             $$

\item{d)}  By re-expressing the Hamiltonian of part b)
           in terms of the mechanical momentum of the
           particle---and by using the results derived
           in part c)---obtain the most general solution
           for ${\pmbf\pi}(t)$ by using Poisson's 
	   equation:
             $$
               \frac{d{\pmbf\pi}}{dt}=[{\pmbf\pi},H] \;.
             $$
             Interpret your results on the basis of a
             ``conventional'' (Newton's second law plus
             Lorentz force) approach (see Problem 10).
\end{description}
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