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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 3}\\%
\vspace {3mm} (January 21, 2011) \\
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\item{(11a)} Assume a bilinear kinetic Energy
$$ T = \sum_{j,k} a_{jk}\,\dot{q}_j\dot{q}_k~~{\rm and\ prove}~~
   \sum_i \dot{q}_i\,\frac{\partial T}{\partial\dot{q}_i} = 2T\ .$$
Due in class (2 points). \bigskip

\item{(11b)} Generalized Momentum:
$${\rm Calculate}~\frac{\partial L}{\partial\dot{x_i}},\
i=1,2,3,~{\rm for}~L=\frac{1}{2}\,m\,\vec{v}^{\,2}-V(\vec{x}).$$
Due in class (2 points). \bigskip

\item{(12)} 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 January 26 before class (10 points).

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