% BB Mar 10  2010
\documentclass[12pt]{article}
\textheight 9.5in
\textwidth  6.0in
\topmargin -0.7in  
\headsep    0.4in  
\oddsidemargin  0.25in  
\evensidemargin 0.25in  

\def\pmbf#1{{\mbox{\boldmath${#1}$}}}
\def\undertext#1{$\underline{\smash{\hbox{#1}}}$}

\pagestyle{empty}
\begin{document}

%----------------------------------------------------%
%%%%%%%%%%%%%%%%% ADVANCED DYNAMICS %%%%%%%%%%%%%%%%
%----------------------------------------------------%
\begin{center}
{\bf MIDTERM ADVANCED DYNAMICS --- PHY--4241/5227}\\
\vspace {1mm}{March 17, 2010, 9:05-9:55 am, HCB 317} \\ 
\end{center}
\setcounter{equation}{0}
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\bf PROBLEM 1 (20 points)}
\medskip
\noindent
\begin{enumerate}
\item Write down the Euler-Lagrange equations and the definition
of the generalized momentum.
\item Write down the Lagrangian of the 1D harmonic oscillator and
continue with this system.
\item Find the momentum $p$ from the Lagrangian.
\item Write down the Hamiltonian $H(p,x)$.
\item Write down Hamilton's equations.
\item Show that Hamilton's equations give Newton's force law.
\end{enumerate}
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%

\centerline{\bf PROBLEM 2 (20 points)}
\begin{enumerate}
\item Use spherical coordinates $(\theta,\phi,r)$ to write down the 
      Lagrangian for a particle of mass $m$ moving in the presence of 
      a spherically symmetric potential $V(r)$, $r=|\vec{r}|$, $\theta$
      polar and $\phi$ azimuth angle. 
\item Write down the Euler-Lagrange equations for the two angles.
\item Use the Lagrangian to identify two conserved quantities.
\item Find $\phi(t)$ for the solution with the initial condition 
      $\dot{\phi}(0)=0$ where $\phi$ is the azimuth angle. 
\end{enumerate}
\hfil\break
%%%%%%%%%%%%%%%%%%%%%%%%%%

\centerline{\bf PROBLEM 3 (10 points)}
\medskip

\noindent
Use SI units and write down the four Maxwell equations (in vacuum) 
in the presence of a charge density $\rho$ and a current density 
$\vec{J}$.

\bigskip
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%

\vfill
\end{document}
