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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 7}\\%
\vspace {3mm} (February 21, 2011) \\
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Read Landau-Lifshitz \textsection 23. Due February 23.

Read Landau-Lifshitz \textsection 24. Due February 25.

Read Landau-Lifshitz \textsection 25. Due March 2.

Read Landau-Lifshitz \textsection 26. Due March 4.

\begin{description}

\item{(21a)} Derive the results of problem~4 \textsection 21 of 
Landau and Lifshitz. Due February 23 before class (3 points).

\item{(21b)} Let $\gamma=\omega+\epsilon$. Taylor expand
$$ x(t) = a\,\cos(\omega\,t+\alpha)+\frac{f\,\left[\cos(\gamma\,t+\beta)
   - \cos(\omega\,t+\beta)\right]}{m\,\left(\omega^2-\gamma^2\right)} $$
to order $\epsilon$ and take the limit $\epsilon\to 0$.
Due in class (3 points).

\item{(22)} Derive the results of problem~1 \textsection 22 of 
Landau and Lifshitz. Due February 25 before class (4 points).

\item{(10b)} Calculate the normal modes (eigenfrequencies) of 
the pendulum of problem 10, set~2. Due February 28 before class 
(4 points).

\item{(10c)} Continue with the double pendulum from assignment~10b.
\begin{enumerate}
  \item Use the eigenfrequencies $\omega_{\pm}$ given in the posted 
   solution of 10b and normal coordinates (Landau-Lifshitz p.67/8) 
   to write down the general solution for the two angles.
  \item Express the integration constants of your solution through 
        the angular positions and velocities at time $t=0$, denoted
        by $\phi_0,\, \dot{\phi}_0,\, \psi_0,\,\dot{\psi}_0$.
  \item Use $\sqrt{l/g}$ as time unit and plot the solutions $\phi(t)$ 
        and $\psi(t)$ up to $t=50\,\sqrt{l/g}$ for initial conditions
        $\phi_0=0,~\dot{\phi}_0=1,~\psi_0=0,~\dot{\psi}_0=-1$. 
\end{enumerate}
Due March 4 before class (10 points).

\item{(21c)} Consider $x(t)=a\,\exp(-\lambda\,t)\,\cos(\omega\,t+\alpha)$. 
Let $\omega=\epsilon$, $a=a_1/\epsilon$, $\alpha=\pi/2+\epsilon/a_2$,
$a_1,\,a_2\ne 0$ and calculate $x(t)$ in the limit $\epsilon\to0$.
Due March 14 before class (3 points).

\end{description}

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