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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 1}\\%
\vspace {3mm} (January 5, 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 (10 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 January 12 before 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 January~7 before 
class.

\item {(4)} Write down your expectations for this course. What are
the main objectives? How do they relate to your present knowledge?
Due January~10 before class. Be detailed (up to 8~points)!

\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 $V(x)=x^2/2$.
\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~10 before class (10 points).

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\item Assignments 5 to 8 are motivated by Feynman's text:
 
\item{(5)} Show that ``the mean square of something that deviates
around an average ... is always greater that the square of the mean''
(4~points). % Due January~10 before class (4~points).
 
\item{(6)} What is a 3D conservative force? Write down two definitions
           (2~points). % Due January~10 in class (2~points).

\item{(7)} Derive the Euler-Lagrange equations from 
$$\delta \int_{t_1}^{t_2} dt\,L = 0\,,\qquad L=L(q_i,\dot q_i,t)\,,
   \qquad i=1,\dots,n_f\,.$$
Due January 10 in class (8 points). 

\item{(8)} Write down and solve the Euler-Lagrange equation for
a free particle, $T=m\dot{x}^2/2$, $U=0$.
Due January 11 in class (2 points).  \medskip
 
\item{(9)} Draw 2D projections when possible.
\begin{enumerate}
\item Write $x,\,y,\,z$ in cylindrical coordinates $\rho,\,\phi,\,z$ 
      (1~point).
\item Write cylindrical coordinates in spherical coordinates (1~point).
\item Write $x,\,y,\,z$ in spherical coordinates $r,\,\theta,\,\phi$ 
      (1~point).
\item 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 $\hat{x},\,
      \hat{y},\,\hat{z}$ (3~points).
\item Make a table of the nine dot products between spherical and
      Cartesian unit vectors (3~points).
\item Express $\vec{v}$ in spherical coordinates using also spherical
      unit vectors (3~points).
\item Express $\vec{v}^{\,2}$ in spherical coordinates (2~points).
\end{enumerate}
Due January~12 before class.

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