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%%%%%%%%%%%%%%%%% ADVANCED DYNAMICS %%%%%%%%%%%%%%%%
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{\bf SOLUTIONS MIDTERM ADVANCED DYNAMICS --- PHY--4241/5227}\\
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\centerline{\bf PROBLEM 1 (20 points)}
\begin{enumerate}
\item $$ \frac{\partial L}{\partial q_i} = \frac{d~}{dt} 
         \frac{\partial L}{\partial \dot{q}_i}\,,~~~p_i = 
         \frac{\partial L}{\partial \dot{q}_i}\,,~~~
         {\rm for}~~~ i=1,\dots,n\,.  $$
\item $$ L = \frac{1}{2}\,m\,\dot{x}^2 - \frac{1}{2}\,k\,x^2\,.$$
\item $$ p = \frac{\partial L}{\partial\dot{x}} = m\,\dot{x}\,.$$
\item $$ H = \frac{p^2}{2m} + \frac{1}{2}\,k\,x^2\,.$$
\item $$ \dot{x} = \frac{\partial H}{\partial p} = \frac{p}{m}\,,
     ~~~-\dot{p} = \frac{\partial H}{\partial x} = k\,x\,. $$
\item Combining the previous two equations we find 
$$ \dot{p} = m\,\ddot{x} = - k\,x\,.$$
\end{enumerate}
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\centerline{\bf PROBLEM 2 (20 points)}
\begin{enumerate}
\item $$L = \frac{1}{2}\,m\,\dot{r}^2 + \frac{1}{2}\,m\,r^2\,\left(
      \dot{\theta}^2 + \sin^2(\theta)\,\dot{\phi}^2\right) - V(r)\,.$$
\item $$0 = \frac{d~}{dt}\frac{\partial L}{\partial\dot{\theta}} -
      \frac{\partial L}{\partial\theta} = 2m\,r\,\dot{r}\,\dot{\theta}
      + m\,r^2\,\left(\ddot{\theta}-\sin(\theta)\,\cos(\theta)\,
      \dot{\theta}\,\dot{\phi}^2\right)\,, $$
      $$0 = \frac{d~}{dt}\frac{\partial L}{\partial\dot{\phi}} -
      \frac{\partial L}{\partial\phi} = \frac{1}{2}\,\frac{d~}{dt}
      \left(m\,r^2\,\sin^2(\theta)\,\dot{\phi}\right)\,.$$
\item From the last equation we find that angular momentum is conserved:
      $$ L_{\phi} = m\,r^2\,\sin^2(\theta)\,\dot{\phi} =
         {\rm constant}\,.$$
      As the Lagrangian $L$ does not explicitly depend on time, energy
      $$ E = \sum_i \frac{\partial L}{\partial\dot{q}_i}\,\dot{q}_i -L$$
      is conserved. Here
      $$E = \frac{1}{2}\,m\,\dot{r}^2 + \frac{1}{2}\,m\,r^2\,\left(
      \dot{\theta}^2 + \sin^2(\theta)\,\dot{\phi}^2\right) + V(r)\,.$$
\item Let $\dot{\phi}(0)=0$ and $\theta\ne 0,\,\pi$. Then for {\it all}  
      times 
      $$\sin^2[\theta(t)]\,\dot{\phi}(t) = 0\ \Rightarrow\
      \dot{\phi}(t) = 0\ \Rightarrow\ \phi(t) = {\rm constant}\,.$$
      The motion is in a plane.
\end{enumerate}
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\centerline{\bf PROBLEM 3 (10 points)}
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$$ \nabla\cdot\vec{E} = \frac{\rho}{\epsilon_0}\,,\qquad
   \nabla\cdot\vec{B} = 0\,, $$
$$ \nabla\times\vec{E} = -\frac{\partial\vec{B}}{\partial t}\,,\qquad
   \nabla\times\vec{B} = \mu_0\,\vec{J} + \mu_0\,\epsilon_0\,
   \frac{\partial\vec{E}}{\partial t}\,. $$
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