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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 2}\\%
\vspace {3mm} (January 7, 2011) \\
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\item{(6)} Expand to order $\epsilon$ (each one point):
\begin{enumerate}
\item $f(x+\epsilon)$.
\item $f(x_1+\epsilon_1,\dots,x_n+\epsilon_n)$.
\item $f(v_1+\epsilon_1,\dots,v_n+\epsilon_n)$.
\item $v'^2=v^2(v_1+\epsilon_1,v_2+\epsilon_2,v_3+\epsilon_3)$ with
      $v^2=v^2(v_1,v_2,v_3)$.
\item $f(v'^2)$.
\end{enumerate}
Due January 10 before class (up to 5 points).

\item Assignments 7 and 8 are motivated by Feynman's text:
 
\item{(7)} Show that ``the mean square of something that deviates
around an average ... is always greater that the square of the mean''
(4~points). Due in class (4~points).
 
\item{(8)} What is a 3D conservative force? Write down two definitions
           (2~points). Due in class (2~points).

\item{(9)} Derive the Euler-Lagrange equations for 
$$\delta \int_{t_1}^{t_2} dt\,L = 0\,,\qquad L=L(q_i,\dot q_i,t)\,,
   \qquad i=1,\dots,s\,.$$
Due in class (8 points). 

\item{--} Read Landau-Lifshitz \textsection 6, \textsection 7, 
\textsection 8. Due January 14 before class.

\item{(10)} 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 $\phi,\,
        \dot{\phi},\,\psi,\,\dot{\psi}$.)
\end{enumerate}
Due January 19 before class (10 points).

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