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%%%%%%%%%%%%%%%%% CLASSICAL MECHANICS %%%%%%%%%%%%%%%%
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\begin{center}
{\bf ADVANCED DYNAMICS --- PHY--4241}\\
\vspace {2mm} {\bf Final Exam}\\
\vspace {1mm}{April 28, 2003} \\ 
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\noindent
The final exam consists of two problems. You must select one
and only one of them; no extra (nor partial) credit will be
given for solving both. You have exactly two hours to complete
the exam. You may {\bf NOT} consult any books or notes; you
will only need pencil and paper to solve the exam. Please 
write clearly! {\bf Best of luck and have a great summer}.

\medskip
\noindent
{\bf Academic Honor Code:}
{\it Students are expected to uphold the Academic Honor Code 
published in the Florida State University Bulletin and the 
Student Handbook. The first paragraph reads: The Academic Honor 
System of Florida State University is based on the premise that 
each student has the responsibility (1) to uphold the highest 
standards of academic integrity in the student's own work, (2) 
to refuse to tolerate violations of academic integrity in the 
University community, and (3) to foster a high sense of integrity 
and social responsibility on the part of the University community.}
\bigskip

\noindent{\bf Student's Name}:\\
\noindent{\bf Student's Social Security Number:}\\
\vspace{1in}

\noindent{$\overline{\bf Student's \ Signature}$}\\
I have upheld the Academic Honor Code.
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\centerline{\bf PROBLEM 1}

\noindent
An electron (of charge $e$) is traveling at a constant
velocity ${\bf v}=v{\hat{\bf x}}$ along the x-axis with
respect to the laboratory frame. You may assume that at 
time $t\!=\!0$ the electron passes through the origin of 
the laboratory frame. 

\begin{description}
\item{(a)}
 Compute the electric ${\bf E}'({\bf r}',t')$  and magnetic 
 ${\bf B}'({\bf r}',t')$ fields in the rest frame of the 
 electron, {\it i.e.,} as measured by an observer in an 
 inertial frame traveling with the electron.

\item{(b)}
 From the above configuration of fields, construct the 
 electromagnetic field tensor $F'_{\mu\nu}({\bf r}',t')$
 as measured by an observer in an inertial frame traveling 
 with the electron.

\item{(c)}
 Use the appropriate Lorentz transformation on the
 electromagnetic field tensor to extract the electric 
 ${\bf E}({\bf r}_{0},t)$ and magnetic 
 ${\bf B}({\bf r}_0,t)$ fields measured by an observer 
 in the laboratory frame located at the point
 ${\bf r}_{0}\!=\!(0,y_{0},0)$ (see the figure).
\end{description}
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\centerline{\bf PROBLEM 2}

\noindent
Three identical mass points of mass $M$, constrained 
to move in a circle, are connected to each other and 
to a fixed support by means of four identical springs
of spring constant $k$, as indicated in the figure.

\begin{description}
\item{(a)} 
 Obtain the Lagrangian of the system in the limit of 
 small oscillations. That is, compute the $3\times 3$ 
 symmetric matrices $\hat{\mathcal T}$ and $\hat{\mathcal V}$
 such that the Lagrangian may be written as
 $$
  L=\frac{1}{2}\dot{\eta}^{T}\hat{\mathcal T}\dot{\eta}
   -\frac{1}{2}\eta^{T} \hat{\mathcal V}\eta\;,
   \quad \Big(\eta^{T}\equiv(x_1,x_2,x_3)\Big)\;,
 $$
 where $x_{n}$ is the displacement of the $n$th mass 
 point from to its equilibrium position.

\item{(b)} 
 Obtain the normal frequencies (eigenvalues) and normal
 modes (eigenvectors) of the problem. Note that there
 is no need to normalize the eigenvectors.

\item{(c)} 
 Assuming that particle number 1 is displaced by one
 unit from its equilibrium position at time $t\!=\!0$
 (that is, $x_1(0)\!=\!1$ and $x_2(0)\!=\!x_3(0)\!=\!
 \dot{x}_1(0)\!=\!\dot{x}_2(0)\!=\!\dot{x}_3(0)\!=\!0$)
 compute the coordinates of all three particles as a
 function of time.
\end{description}
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