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{\bf Intermediate Mechanics II --- PHY 4936}\\\vspace {2mm} 
{\bf Midterm Exam October 7, 2011} \\ 
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\centerline{\bf PROBLEM 1 (33 points)}
\noindent
The potential of the 1D harmonic oscillator is $U(x)=k\,x^2/2$.
\begin{enumerate}
\item Write down the Lagrangian.
\item Write down the Euler-Lagrange equation.
\item Solve the Euler-Lagrange equation. Express integration
constants through time zero initial conditions $x_0$ and $\dot x_0$.
\end{enumerate}
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\centerline{\bf PROBLEM 2 (34 points)}
\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 Identify the conservation laws.
  \item Find the special solutions for $\theta = {\rm constant.}$
        Describe this motion.
\end{enumerate}
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\centerline{\bf PROBLEM 3 (33 points)}
\medskip \noindent
Assume a Lagrangian $L=L(\{q_i\},\{\dot{q}_i\},t)$ where $q_i,\,i=1,
\dots,n$ are generalized coordinates, $\dot{q}_i,\,i=1,\dots,n$ are 
generalized velocities and $t$ is the time.
\begin{enumerate}
\item Write down the principle of least action.
\item Derive the Euler-Lagrange equations from the principle of least 
      action.
\item Assume that the Lagrangian is invariant under translation
      $q_k\to q_k'=q_k+\epsilon_k$ of one or more generalized
      coordinates $q_k$. Find the corresponding conserved quantities.
\end{enumerate}
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