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{\bf ADVANCED DYNAMICS --- PHY--4241/5227}\\\vspace {2mm} 
{\bf Midterm Exam March 21, 2011} \\ 
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{\bf Each problem counts 25 points.}
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\centerline{\bf PROBLEM 1}
\noindent
Write the principle of least action and the 
Euler-Lagrange equation(s) of motion for a 
1-dimensional Lagrangian of the form:
$$
  L=\frac{1}{2}m\dot{x}^2-V(x) \;.
$$
Is the resulting equation consistent with 
Newton's second law?
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\centerline{\bf PROBLEM 2}
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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 Use the definition of the generalized momentum to 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 3}
\noindent
Consider the 3-dimensional Lagrangian for a particle
of mass $m$ moving in the presence of a spherically
symmetric potential $V({\bf r})\!=\!V(r)$. That is,
$$
   L=\frac{1}{2}m\Big(\dot{r}^{2}+
            r^{2}\dot{\theta}^{2}+
            r^{2}\sin^{2}\theta\dot{\phi}^{2}\Big)
            -V(r)\;.
$$
Identify as many conserved quantities ({\it i.e.,} 
constants of the motion) as you can from the mere 
structure of the Lagrangian.
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\centerline{\bf PROBLEM 4}
\noindent
Consider a point mass $m$ on the surface of a sphere of radius $R$ 
under the influence of gravity $-g\hat{z}$ (spherical pendulum).
\begin{enumerate}
  \item Write down the Lagrange function using spherical coordinates.
  \item Find the Euler-Lagrange equations.
  \item Calculate the special solutions for $\theta = {\rm constant.}$
        Describe this motion.
\end{enumerate}
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