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%%%%%%%%%%%%%%%%% ADVANCED DYNAMICS %%%%%%%%%%%%%%%%
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{\bf MAKE UP MIDTERM ADVANCED DYNAMICS --- PHY--4241/5227}\\
% \vspace {1mm}{March 17, 2010} \\ 
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\centerline{\bf PROBLEM 1 (20 points)}
\medskip
\noindent
Let the interaction of two point particles be described by the 
Lagrangian
$$ L = \frac{m_1}{2}\dot{\vec{r}}_1^{\,2}
     + \frac{m_2}{2}\dot{\vec{r}}_2^{\,2}
     - \frac{k}{2}\,\left(\vec{r}_1-\vec{r}_2\right)^2\,. $$
\begin{enumerate}
\item Is the energy of this system conserved (with reason)?
\item Write down the definition of the center of mass $\vec{R}(t)$ 
      of the system. How is the center of mass reference frame
      defined?
\item Let $\vec{r}=\vec{r}_1-\vec{r}_2$. Express the Lagrangian
in the center of mass frame as function of the coordinate
$\vec{r}$ and velocity $\dot{\vec{r}}$.
\item Use Cartesian coordinates and write down the Euler-Lagrange
      equations of the system.
\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 (10 points)}
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Shortly explain the principle behind each definition. Precise
numbers are not important.
\begin{enumerate}
\item How is one second [s] defined?
\item How is one meter [m] defined?
\item How is one kilogram [kg] defined?
\item How is one Newton [N] defined?
\item How is one Ampere [A] defined?
\item How is one Coulomb [C] defined?
\end{enumerate}

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