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{\bf FINAL ADVANCED DYNAMICS -- PHY 4241/5227\ (April 29, 2009)}\\ 
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\centerline{\bf PROBLEM 1 (30 points)}
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\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 translations
      $q_i\to q_i'=q_i+\epsilon_i$. Find the corresponding conserved
      quantities.
\end{enumerate}
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\centerline{\bf PROBLEM 2 (30 points)}

An electromagnetic field tensor (in arbitrary units) is
given by
$$ (F^{\alpha\beta}) = \left( \matrix{ 
			      \ 0& \ 1& \ 2&  -1 \cr
			       -1& \ 0&  -2& \ 1 \cr
			       -2& \ 2& \ 0& \ 3 \cr
			      \ 1&  -1&  -3& \ 0 \cr}\right) \ . $$
Find (the Einstein summation convention is assumed for identical
indices):
\begin{enumerate}
\item $F^{\alpha}_{\ \ \alpha}$. 
\item The matrix $(F_{\alpha\beta})$.
\item $F_{\alpha\beta} F^{\alpha\beta}$.
\end{enumerate}
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\centerline{\bf PROBLEM 3 (40 points)}
\begin{enumerate}
\item Write down the three components, $E^1$, $E^2$ and $E^3$ of the
      electric field of a static point charge $q$, which is located
      at the origin of a coordinate system described by $x^0$, $x^1$,
      $x^2$ and $x^3$.
\item What is the magnetic field $\vec{B}$ of this point charge?      
\item Write the down the electromagnetic field tensor $(F^{\alpha\beta})$
      of this point charge.
\item Use the transformation law $F'(x)=A\,F(x)\,A^{\dagger}$ and the 
      Lorentz matrix
      $$ A\ =\ \left( \matrix{ \gamma & -\gamma\beta & 0 & 0 \cr
                         -\gamma\beta &  \gamma      & 0 & 0 \cr 
                               0      &       0      & 1 & 0 \cr
                               0 & 0 & 0 & 1 \cr} \right) $$
     to calculate the electromagnetic field tensor $(F'^{\alpha\beta})$
     for an inertial frame $K'$, which moves with velocity $v$ along the
     $x$-axis of the rest frame of the point charge (the orientations
     of the axis of the two frames are parallel and their origins 
     coincide at time $t=t'=0$).
\item The previous expression $F'(x)$ is still written in $x$-coordinates. 
      Use now $x'$-coordinates to write down the electric $E'^i(x')$ 
      and magnetic $B'^i(x')$, $i=1,2,3$ field components in the frame 
      $K'$.
\end{enumerate}
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