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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 2}\\%
\vspace {3mm} (January 12, 2009) \\
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\item{(8a)} 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 14 in class (6 points). 

\item{(8b)} Write down and solve the Euler-Lagrange equation for
a free particle, $T=m\dot{x}^2/2$, $U=0$.
Due January 14 in class (2 points).  \medskip
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\item{(9)} 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 An unphysical system with $T=0$, $U=-x^2/2$. Is the action a mimimum?
\item An unphysical system with $T=0$, $U=x$.
\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 16 in class (6 points).  \medskip
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\item{(10)} 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 26 in class (6 points).
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\item{(11)} A double pendulum consists of two simple pendula, with 
one pendulum suspended from the bob of the other. Assume that the two 
pendula have equal lengths, have bobs of equal mass and are confined 
to move in the same plane. 
\begin{enumerate}
  \item Define angles $\phi$ and $\psi$ for the pendula with respect 
        to the gravity direction and write down the Lagrange function.
  \item Derive the  equations of motion for small oscillations 
        around the rest position $\phi = \psi =0$.
  \item Calculate the normal modes.
\end{enumerate}
Due January 26 before class (10 points).
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