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{\bf ADVANCED DYNAMICS --- PHY 4936}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 5}\\%
\vspace {3mm} (October 13, 2011) \\
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Read Landau-Lifshitz up to p.39 (\textsection 15).

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\item{(19)} Consider the effective potential of the Kepler problem
$$ U_{\rm eff}(r) = - \frac{\alpha}{r} + \frac{L^2}{2\,m\,r^2}\ . $$

\noindent (A) Calculate the value $r_0$ where $U_{\rm eff}(r)$ 
          has its minimum.

\noindent (B) Calculate $U_{\rm eff}^{\min}=U_{\rm eff}(r_0)$.

\noindent (C) Assume an energy $E<0$ and calculate the values of
          the turning points $r_{\min}$ and $r_{\max}$.

\noindent (D) For which energy range do we have two real solutions?

Due in class, 5 points (2 for C, 1 for A, B, D).

\item{(20)} Assume $0<e<1$ for the eccentricity and transform the
elliptic equation 
$$ \frac{p}{r} = 1 + e\,\cos(\phi)\,,~~p>0~~{\rm into\ the\ form}
 ~~\frac{x'^2}{a^2}+\frac{y'2}{b^2} = 1\ .$$ 
This means, {\it derive} the definitions of $x'$, $y'$, major half-axis
$a$ and minor half-axis $b$ in terms of $x$, $y$, $p$ and~$e$. Due 
Wednesday, October 19 before class, 8~points.

\item{(21)} Use the inertial frame of \textsection 15 of the book and
plot the orbits for the four initial conditions of the table in set~4
(best in one figure). Indicate the initial positions. Due October~24
before class, 12 points. 

\item{(22)} Calculate the turning point $\theta_{\min}$ and 
$\theta_{\max}$ for the spherical pendulum of the midterm for 
the special case $mgR = M^2_z/(4mR^2) = E/2$. Due in class (6 points).

Read Chapter V of Landau-Lifshitz up to \textsection 26, p.79.

\item{(23)} Derive the results of problem~3 \textsection 21 of 
Landau and Lifshitz (frequency of oscillations). Due October 26 
before class (4 points).

\item{(24)} Calculate $a$ and $\alpha$ of Eq.~(22.4 and 1/2) of Landau
and Lifshitz in terms of $a$ and $\alpha$ of Eq.(22.4) of Landau and
Lifshitz.  Then, 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 (4 points). Now October 28 before class.

\item{(25)} Calculate the normal modes (eigenfrequencies) of 
the pendulum of problem 8, set~2. Due October 31 before class 
(4 points).

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