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\begin{center}
{\bf ADVANCED DYNAMICS --- PHY 4936}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 6}\\%
\vspace {3mm} (November 1, 2011) \\
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Read Landau-Lifshitz p.58 up to p.72 (\textsection 21 to 
\textsection 24).

\begin{description}

\item{(26)} Continue with the double pendulum from assignment~25.
\begin{enumerate}
  \item Use the eigenfrequencies $\omega_{\pm}$ given in the posted 
   solution of 25 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
        $\phi_0=0,~\dot{\phi}_0=\sqrt{g/l},~\psi_0=0,~\dot{\psi}_0=
        -\sqrt{g/l}$. 
\end{enumerate}
Due November 7 before class (10 points).

\item{(27)} (A) Calculate the eigenfrequencies of a 2D harmonic oscillator
$$ \left(\sum_{k=1}^2 m_{ik}\,\ddot x_k + k_{ik}\,x_k
   \right)\ =\ 0\,,~~(i=1,2) $$ 
with matrix elements
$$ M\ =\ \left(m_{ik}\right)\ =\ \left(\matrix{1&1\cr 1&2 \cr}\right)
~~{\rm and}~~
   K\ =\ \left(k_{ik}\right)\ =\ \left(\matrix{5&1\cr 1&2 \cr}\right)\,. $$
Due in class (4 points).

(B) Use normal co-ordinates $\Theta_1$ and $\Theta_2$ as defined in 
Landau-Lifshitz (p.67/68). Express $x_1$ and $x_2$ in terms of them.
Due in class (4 points).

(C) Use the given numbers for $m_{ik}$ and $k_{ik}$ and write down
the Lagrangian as function of $\dot x_1$, $\dot x_2$, $x_1$ and $x_2$.
Then, substitute normal co-ordinates as found in (B) and write down the
Lagrangian in terms of $\dot \Theta_1$, $\dot \Theta_2$, $\Theta_1$ 
and $\Theta_2$. Due November 7 before class (6 points).

(D) Assume at time $t=0$ the initial conditions $\Theta_1(0)=1$, 
$\dot\Theta_1(0)=0$, $\Theta_2(0)=0$ and $\dot\Theta_2(0)=1$. Plot 
the resulting solution first in in the $\Theta_1$-$\Theta_2$ plane 
and then in the $x_1$-$x_2$ plane. Due November 9 before class 
(4 points).

\item{(28)} The Lagrangian of a 2D oscillator is
$$ L\ =\ \frac{m}{2}\,\left( \dot x^2 + \dot y^2 
      - \omega_1^2\,x^2 - \omega_2^2\,y^2 \right)\ .  $$
Write down the general solution for the case that $x=y=0$ at $t=0$.
Which condition applies to $\omega_1$ and $\omega_2$, so that the
mass point returns to $x=y=0$ at some future time $t$? How long
will it take?  Due November 14 before class (4 points).

\end{description}

Read Landau-Lifshitz p.96 up to p.101 (\textsection 31 and
\textsection 32).

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