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\begin{center}
{\bf ADVANCED DYNAMICS --- PHY--4241/5227}\\\vspace {2mm} 
{\bf HOMEWORK 12}\\%
\vspace {3mm} (\today) \\
\vspace {1mm}{Due on Monday, April 21, 2003} \\ 
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\centerline{\bf PROBLEM 35}
\centerline{({\small Marion and Thornton Problem 12.3})}

\noindent
Two identical harmonic oscillators (with masses M and natural
frequencies $\omega_{0}$) are coupled such that by adding to 
the system a mass $m$ common to both oscillators the equations 
of motions become
\begin{eqnarray}
&& \ddot{x}_{1}+(m/M)\ddot{x}_{2}+\omega_{0}^{2}x_{1}=0 \nonumber \\
&& \ddot{x}_{2}+(m/M)\ddot{x}_{1}+\omega_{0}^{2}x_{2}=0 \;. \nonumber
\end{eqnarray}
Solve these coupled equations, and obtain the frequencies 
{\bf and normal modes} of the system. 
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\centerline{\bf PROBLEM 36}
\centerline{({\small Marion and Thornton Problem 12.21})}

\noindent
Three oscillators of equal mass $m$ are coupled such that the
potential energy of the system~is 
$$
  U=\frac{1}{2}\Big[\kappa_{1}(x_{1}^{2}+x_{3}^{2})+
                    \kappa_{2}x_{2}^{2}+
		    \kappa_{3}(x_{1}x_{2}+x_{2}x_{3})\Big]\;,
$$
where $\kappa_{3}=\sqrt{2\kappa_{1}\kappa_{2}}$. Find the
eigenfrequencies {\bf and eigenvectors} by solving the 
secular equation. What is the physical interpretation
of the zero-frequency mode?
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\centerline{\bf PROBLEM 37}
\centerline{({\small similar to Marion and Thornton Problem 12.18})}

\noindent
A simple plane pendulum of mass $m$ and length $l$ hangs
from a block of mass $M$ which is constrained to slide on
a smooth horizontal track as shown in the figure.
\begin{description}
\item{a)} Assuming that $\theta\!\ll\!1$, obtain the
          Lagrangian of the system in the limit of
          small oscillations. That is, compute
          the $2\times 2$ symmetric matrices
          $\hat{\mathcal T}$ and $\hat{\mathcal V}$
          such that the Lagrangian may be written as
          $$
           L=\frac{1}{2}\dot{\eta}^{T}
             \hat{\mathcal T}\dot{\eta}
            -\frac{1}{2}\eta^{T}
             \hat{\mathcal V}\eta\;,
	     \quad \Big(\eta^{T}\equiv(x,z)\Big)\;.
          $$

\item{b)} Find the normal modes and the associated
          frequencies of small oscillations of the
          system.

\item{c)} Given the following set of initial conditions:
          $z(0)\!=\!z_{0}$ and
          $x(0)\!=\!\dot{x}(0)\!=\!\dot{z}(0)\!=\!0$,
          predict the time-dependent motion of the system.
          That is, compute $x(t)$ and $z(t)$ for all times.
\end{description}
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