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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 2}\\%
\vspace {3mm} (January 12, 2010) \\
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\begin{description}

\item{(9)} Read Chapter~6.1 to 6.3 of M\&T. Due January 20 before class.

\item{(10)} 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~15 before class (10 points).

\item{(11)} Write down and solve the Euler-Lagrange equations for
\begin{enumerate}
\item The one-dimensional harmonic oscillator $T=m\dot{x}^2/2$, $U=k\,x^2/2$.
\item An unphysical system with $T=0$, $U=x^2/2$. Is the action a minimum?
\item The three-dimensional harmonic oscillator $T=m\vec{v}^{\,2}/2$, 
$U=k\,\vec{r}^{\,2}/2$.
\item Circular motion without friction $T=m\,l^2\,{\dot\theta}^{\,2}/2$, 
      $U={\rm const}$.
\end{enumerate}
Due January 13 and 15 in class (8 points).  \medskip

\item{(12)} Consider the spherical pendulum (a point mass $m$ on 
the surface of a sphere of radius $R$ under the influence of gravity 
$-g\hat{z}$).
\begin{enumerate}
  \item Write down the Lagrange function using spherical coordinates.
  \item Find the Euler-Lagrange equations.
  \item Calculate the special solutions for $\theta = {\rm constant.}$
        Describe this motion.
\end{enumerate}
Due January 22 before class (10 points).

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