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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 4}\\%
\vspace {3mm} (January 27, 2010) \\
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\begin{description}

\item{(16)} Calculate explicitly
$$ \delta_x L\ {\rm for\ the\ Lagrangian\ of\
two\ harmonically\ bound\ particles\ in\ 1D}$$
$$L=\frac{m_1}{2}(\dot{{x}}_1)^2+\frac{m_2}{2}(\dot{x}_2)^2
   -k\,(x_1-x_2)^2\ .$$
Is there a conservation law? Due February 1 in class (2 points).

\item{(17)} Continue with the double pendulum from assignment~13.
\begin{enumerate}
  \item Use eigenvalues $\omega_{\pm}$ as given in the posted solution
        and calculate corresponding eigenvectors in the form
         $$ \left(\matrix{1 \cr \psi_{0\pm}} \right)\ . $$
  \item Are these eigenvectors orthogonal?
  \item Write down the general solution.
  \item Express the integration constants of your solution through 
        the angular positions and velocities at time $t=0$, denoted
        by $\phi_0,\, \dot{\phi}_0,\, \psi_0,\,\dot{\psi}_0$.
  \item Use $\sqrt{l/g}$ as time unit and plot the solutions $\phi(t)$ 
        and $\psi(t)$ up to $t=50\,\sqrt{l/g}$ for initial conditions
        $\phi_0=0,~\dot{\phi}_0=1,~\psi_0=0,~\dot{\psi}_0=-1$. 
\end{enumerate}
Due February 1 before class (10 points).

\item{(18)} Consider the spherical pendulum of mass $m$ again.

(a) Use the Legendre transformation
$$ H(\theta,\phi,p_{\theta},p_{\phi})=
   \dot{\theta}p_{\theta}+\dot{\phi}p_{\phi} -
   L(\theta,\phi,\dot{\theta},\dot{\phi})\;, $$  
to construct the Hamiltonian of the system and show that it is 
identical to the energy $E\!=\!T\!+\!V$.

(b) Write down Hamilton's equations of motions for the
          system and identify a conserved quantity.

Due February 3 before class (6 points).

\item{(19)} Calculate explicitly
$$ \delta_{x^i} L\ (i=1,\dots,3)\ {\rm for\ the\ 
Lagrangian\ of\ the\ 3D\ Kepler\ problem} $$
$$L=\frac{m_1}{2}(\dot{\vec{x}}_1)^2+\frac{m_2}{2}(\dot{\vec{x}}_2)^2
   -\frac{g\,m_1m_2}{\left|\vec{x}_1-\vec{x}_2\right|}$$
Are there associated conservation laws? 
Due February 3 in class (4 points).

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