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\begin{center}
{\bf ADVANCED DYNAMICS --- PHY--4241/5227}\\\vspace {2mm} 
{\bf First Midterm Exam}\\%
\vspace {1mm}{February 17, 2003} \\ 
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\centerline{\bf PROBLEM 1}
\noindent
Write the principle of least action and the 
Euler-Lagrange equation(s) of motion for a 
1-dimensional Lagrangian of the form:
$$
  L=\frac{1}{2}m\dot{x}^2-V(x) \;.
$$
Is the resulting equation consistent with 
Newton's second law?
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\centerline{\bf PROBLEM 2}
\noindent
Consider the 3-dimensional Lagrangian for a particle
of mass $m$ moving in the presence of a spherically
symmetric potential $V({\bf r})\!=\!V(r)$. That is,
$$
   L=\frac{1}{2}m\Big(\dot{r}^{2}+
            r^{2}\dot{\theta}^{2}+
            r^{2}\sin^{2}\theta\dot{\phi}^{2}\Big)
            -V(r)\;.
$$
Identify as many conserved quantities ({\it i.e.,} 
constants of the motion) as you can from the mere 
structure of the Lagrangian.
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\centerline{\bf PROBLEM 3}
\noindent 
The Hamiltonian for a one-dimensional harmonic 
oscillator of mass $m\!=\!1$ and spring constant 
$k\!=\!1$ is given by the following expression:
$$
    H=\frac{p^{2}}{2}+\frac{x^{2}}{2} \;.
$$
Compute the Poisson bracket of the function
$f(x,p;t)\!=\!x^{2}$ with the Hamiltonian. Is
$f(x,p;t)$ a constant of the motion? Explain.
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\centerline{\bf PROBLEM 4}
\noindent
Write the four Maxwell's equations (in vacuum) 
in the presence of a charge density $\rho$ and 
a current density ${\bf J}$ (you may use SI or 
Gaussian units).
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\centerline{\bf PROBLEM 5}
\noindent
Before Maxwell's contribution to electrodynamics, 
Ampere's law (in Gaussian units) was given by:
$$
  \nabla\times{\bf B}=\frac{4\pi}{c}{\bf J}\;.
$$
Introduce Maxwell's correction to Ampere's law and 
discuss briefly why it proved to be such a fundamental 
contribution.
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\centerline{\bf PROBLEM 6}
\noindent 
Discuss briefly the possibility or impossibility of the
following configurations of electric and magnetic fields.
\begin{description}
 \item{} $${\rm a)}\; 
           {\bf E}=\frac{\alpha}{r^{2}}\hat{\bf r}
           \quad {\rm and} \quad {\bf B}=0\;.$$

 \item{} $${\rm b)}\; 
           {\bf B}=\frac{\alpha}{r^{2}}\hat{\bf r}
           \quad {\rm and} \quad {\bf E}=0\;.$$
\end{description}
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