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{\bf PHY 4938 HOME AND CLASS WORK -- SET 9}\\%
\vspace {3mm} (November 26, 2011) \\
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Read Landau-Lifshitz \textsection 40 and \textsection 42. 

\begin{description}

\item{(36-A)} Legendre transformation: Define the Hamiltonian by
$$ H = \left(\sum_j\dot{q}_j\,\frac{\partial L}{\partial\dot{q}_j} 
  -L \right)\ {\rm and\ the\ generalized\ momentum\ by}\
   p_j=\frac{\partial L}{\partial\dot{q}_j}\ .                  $$
Show that the Hamiltonian is a function of $q_j$ and $p_j$ only:
$H=H(q_j,p_j)$. Then derive Hamilton's equations of motion. 
Hint: Calculate $dH$. Due in class (4~points).

\item{(36-B)} Problem 1, Landau-Lifshitz \textsection 40, p.133.
Show first: $H=T+U$.  Due in class (4~points).

\item{(36-C)} Derive Newton's equation from Hamilton's equation
for the Cartesian case. Due in class (2~points).

\item{(37)} Poisson brackets are defined by
$$[g,h] = \sum_k \left( 
  \frac{\partial g}{\partial q_k}\,\frac{\partial h}{\partial p_k}- 
  \frac{\partial h}{\partial q_k}\,\frac{\partial g}{\partial p_k}
  \right) $$
where $g$ and $h$ are functions of $q_i$, $p_i$ and, possibly, $t$.
Show the following properties (due December~7 before class, 10 points): 
\begin{enumerate}
\item $$ \frac{dg}{dt} = [g,H]+\frac{\partial g}{\partial t}\ .$$
\item $$ \dot{q}_j = [q_j,H]\ .$$
\item $$ \dot{p}_j = [p_j,H]\ .$$
\item $$ [x_{i},x_{j}]=[p_{i},p_{j}]=0\;; \quad
         [x_{i},p_{j}]=\delta_{ij} \;, $$
\item $$ [x_{i},L_{j}] = \epsilon_{ijk}x_k\;,\; 
         [p_{i},L_{j}] = \epsilon_{ijk}p_k\;,\; {\rm and}\;\;  
         [L_{i},L_{j}] = \epsilon_{ijk}L_k\;,\; \;, $$
          where the Einstein summation convention is used and 
          $L_j=\epsilon_{jkl}\,x_k\,p_l$ is the $j$th component
          of the angular momentum of the system.
\end{enumerate}

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\item{(38)} 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 $\rho(q_1,\dots,q_n,p_1,
\dots,p_n,t)$. If their are no sources or sinks, we have a conserved
current
\begin{equation} \label{cc}
  \frac{\partial\rho}{\partial t}+\nabla\cdot\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 in class (6 points). 

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