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\begin{center}
{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 5}\\%
\vspace {3mm} (January 30, 2009) \\
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\begin{description}

\item{(20)} Calculate
$$ \delta_x L\ {\rm for\ the\ Lagrangian\ of\
two\ harmonically\ bound\ particles\ in\ 1D}$$
$$L=\frac{m_1}{2}(\dot{{x}}_1)^2+\frac{m_2}{2}(\dot{x}_2)^2
   -k\,(x_1-x_2)^2\ .$$
Is there a conservation law? Due Februar 2 in class (2 points).

\item{(21)} Calculate
$$ \delta_{x^i} L\ (i=1,\dots,3)\ {\rm for\ the\ 
Lagrangian\ of\ the\ 3D\ Kepler\ problem} $$
$$L=\frac{m_1}{2}(\dot{\vec{x}}_1)^2+\frac{m_2}{2}(\dot{\vec{x}}_2)^2
   -\frac{g\,m_1m_2}{\left|\vec{x}_1-\vec{x}_2\right|}$$
Are there any associated conservation laws? Due February 4 in class 
(4 points).

\end{description}

Read M\&T chapter 7.10 (due February~4).

\begin{description} 
\item{(22)} Consider the spherical pendulum of mass $m$ again and 

(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~9 before class (8 points).

\item{(23)} Liouville's Theorem: We define the velocity in phase space
as a $2n$-dimensional vector $\vec{v}=(\dot{q}_1,\dots,\dot{q}_n,
\dot{p}_1,\dots,\dot{p}_n)$. A large collection of particles can be 
described by their density in phase space $\rho(q_1,\dots,q_n,p_1,
\dots,p_n)$. If their are no sources or sinks, we have a conserved
current
\begin{equation} \label{cc}
  \frac{\partial\rho}{\partial t}+\nabla\left(\vec{v}\,\rho\right)=0
\end{equation} 
$$ {\rm where}\ \nabla=\left(\frac{\partial\ }{\partial q_1},\dots,
  \frac{\partial\ }{\partial q_n},\frac{\partial\ }{\partial p_1},
  \dots, \frac{\partial\ }{\partial p_n}\right)\ {\rm is\ the\ gradient
  \ in\ phase\ space.}  $$
(a) Expand (\ref{cc}) in sums of partial derivatives (you get five
terms when you keep coordinates and momenta in separate contributions).
(b) Use Hamilton's equations of motion to show that two terms cancel 
out. (c) Combine the remaining terms to
\begin{equation} 
   \frac{d\rho}{dt} = 0~~{\rm (Liouville's\ Theorem).} 
\end{equation} 
Due February~6 in class (6 points). Read M\&T chapter 7.12 (due 
February~9).
\end{description}

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