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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 1}\\%
\vspace {3mm} (January 7, 2009) \\
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\item {(1)} Consider light passing from medium~1 into medium~2. Use 
Fermat's principle to derive a relationship between the velocities of 
light in the media, $v_1$ and $v_2$, and the angles of refraction,
$\theta_1$ and $\theta_2$ (a figure will be given in class). 
Due January~7 in class (10 points).
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\item{(2)} Read the Handout, the first nine pages of ``The Principle 
of Least Action'' from Chapter 19 of {\it ``The Feynman Lectures on 
Physics'', Vol. II}. Due January~9 before class.
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\item Assignments 3 and 4 are motivated by Feynman's text:
 
\item{(3)} Show that ``the mean square of something that deviates
around an average ... is always greater that the square of the mean''.
Due January~9 in class (4 points).
 
\item{(4)} What is a conservative force? Write down a definition.
           Due January~9 in class (1 point).

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\item{(5)} Read Chapter~6.1 to 6.3 of M\&T. Due January 12 before class.
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\item{(6)} Write $x,\,y,\,z$ in
\begin{enumerate}
\item Cylindrical coordinates $\rho,\,\phi,\,z$.
\item Spherical coordinates $r,\,\theta,\,\phi$ (define
      $\phi$ as before).
\item Write cylindrical coordinates in terms of spherical coordinates.
\end{enumerate}
Due January~12 in class (3 points).

If you have nothing more to do: Express unit vectors
$\hat{\rho},\,\hat{\phi}$ for cylindrical and $\hat{r},\,
\hat{\theta},\,\hat{\phi}$ for spherical coordinates in terms 
of the cartesian unit vectors $\hat{x},\,\hat{y},\,\hat{z}$.
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\item{(7)} Consider a particle of mass $m=1$, moving from 
$x_1=0$ at time $t_1=0$ to $x_2=1$ at time $t_2=\pi/2$, under 
the influence of a one-dimensional harmonic potential 
of the form $V(x)=x^2/2$.
\begin{enumerate}
\item Using Newton's equations of motion, obtain the time-dependent 
      motion of the system; {\it i.e.,} solve for $x(t)$. Compute 
      the action for this exact path.

\item Using an approximate linear path of the form $x(t)\!=\!a+bt$, 
      compute the action for this path and compare it with the 
      value obtained before. Hint: Make sure that the path is 
      consistent with the boundary conditions.

\item Assume that the action result of (2.) is in units $J\cdot s$ and 
      express it in multiples of $\hbar=1.05\times 10^{-34}\,J\cdot s$.
\end{enumerate}
Due January~14 before class (10 points).
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