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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 1}\\%
\vspace {3mm} (January 6, 2010) \\
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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~6 in class (10 points).

\item {(2)} Write down your expectations for this course. What are
the main objectives? Due January~11 before class (up to 8~points).

\item{(3)} 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~8 before class.

\item Assignments 4 to 7 are motivated by Feynman's text:
 
\item{(4)} 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~11 before class (4~points).
 
\item{(5)} What is a 3D conservative force? Write down two definitions
           (2~points). % Due January~11 in class (2~points).

\item{(6)} 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 11 in class (8 points). 

\item{(7)} 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{(8)} 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~22 before class.

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