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{\bf INTERMEDIATE MECHANICS II --- PHY 4936}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 1}\\%
\vspace {3mm} (August 27, 2011) \\
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\item {(1)} Consider light passing from medium~1 into medium~2. Use 
Fermat's principle to derive Snell's law between the velocities of 
light, $v_1$ and $v_2$, in two media and the angles of refraction,
$\theta_1$ and $\theta_2$ (a figure will be given in class). 
Due in class (8 points).
 
\item{(2)} Method of 2D projections for spherical coordinates. Each of 
the following tasks counts one point. Follow the instructions given in 
class (no other methods). 
\begin{enumerate}
\item Write $x,\,y$ in cylindrical coordinates $\rho,\,\phi$.
\item Write $dx^2+dy^2$ in cylindrical coordinates.
\item Write cylindrical coordinates $\rho,\,z$ in spherical 
      coordinates $r,\,\theta$.
\item Write $d\rho^2+dz^2$ in spherical coordinates.
\item Write $dx^2+dy^2+dz^2$ in spherical coordinates.
\item Write $v^2$, the velocity squared, in cartesian coordinates.
\item Write $v^2$, in ccylindrical coordinates.
\item Write $v^2$, in spherical coordinates.
\end{enumerate}
Due in class (up to 8 points).

\item{(3)} Read Landau-Lifshitz up to page~10 and the Handout, the 
first nine pages of ``The Principle of Least Action'' from Chapter 
19 of {\it ``The Feynman Lectures on Physics'', Vol. II}.  Prepare 
questions about anything you do not understand. 
Due August~31 before class.

\item {(4)} What do you remember from Intermediate Mechanics~I.
Write down the most important results (not more than two pages
in average-sized letters). Due August~31 before class (up to 
8~points, depending on whether you hit the most important points
or not).

\item{(5)} 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 $V(x)=x^2/2$.
Due September~7 before class (10 points).
\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}

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