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{\bf MIDTERM ADVANCED DYNAMICS --- PHY--4241/5227}\\
\vspace {1mm}{March 6, 2009} \\ 
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\centerline{\bf PROBLEM 1 (20 points)}
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\noindent
The Lagrangian of the 1D harmonic oscillator is
$$ L={1\over 2}\,m\,\dot{x}^2-{1\over 2}\,k\,x^2\ .  $$
\begin{enumerate}
\item Find the momentum $p$.
\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}
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\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 $V({\bf r})\!=\!V(r)$. 
\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}
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\centerline{\bf PROBLEM 3 (5 points)}
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\noindent
Write down the four Maxwell's equations (in vacuum) in the presence 
of a charge density $\rho$ and a current density ${\bf J}$ (you may 
use SI or Gaussian units).

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\centerline{\bf PROBLEM 4 (15 points)}

\begin{enumerate}
\item Write down the two basic postulates of the Special Theory of 
      Relativity.
\item The lifetime of a pion is $\tau\!=\!2.6\times10^{-8}$~s (with 
      respect to its rest frame). Assume the pion travels in a collider 
      experiment at 99\% of the speed of light with respect to the Lab
      frame. 
      \begin{enumerate}
      \item How long is the observed lifetime of the pion in the Lab 
            frame?
      \item How far does the pion travel in the Lab frame during one
            lifetime?
      \end{enumerate}
\end{enumerate}

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