
\item{(13)} 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. (Compare problem~1 of Landau-Lifshitz with
$l_1=l_2=l$, $\phi_1=\phi$ and $\phi_2=\psi$.) 

\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 $\phi = \psi =0$. (Small oscillations
        neglect all terms in the Taylor expansion of the Lagrangian, 
        which are higher than quadratic in combinations of $\phi,\,
        \dot{\phi},\,\psi,\,\dot{\psi}$.)
  \item Calculate the normal modes. % (eigenvalues).
\end{enumerate}
Due January 25 before class (10 points).

\end{description}

\noindent Read M\&T chapter 7.9 (due January 25). 

\begin{description}

\item{(14)} Derive (due January 25 in class, 8 points)
$$ \frac{d}{dt}\,\left( L - \sum_j \dot{q}_j\,
   \frac{\partial L}{\partial \dot{q}_j} \right) = 0~~{\rm from}~~
   \delta_t L = \frac{\partial L}{\partial t}\,\delta t = 0\ .$$
Assume a bilinear kinetic Energy
$$ T = \sum_{j,k} a_{jk}\,\dot{q}_j\dot{q}_k~~{\rm and\ prove}~~
   \sum_i \dot{q}_i\,\frac{\partial T}{\partial\dot{q}_i} = 2T\ .$$

\item{(15a)} Generalized Momentum:
$${\rm Calculate}~\frac{\partial L}{\partial x_i},\
i=1,2,3,~{\rm for}~L=\frac{1}{2}\,m\,\vec{v}^{\,2}-V(\vec{x}).$$
Due January 29 in class (2 points).

\item{(15b)} 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 29 in class (6 points).

\end{description}

\end{document}

