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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 6}\\%
\vspace {3mm} (February 9, 2011) \\
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Read Landau-Lifshitz \textsection 15. Due February 11.

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\item{(18)} Consider the effective potential of the Kepler problem
$$ U_{\rm eff}(r) = - \frac{\alpha}{r} + \frac{L^2}{2\,\mu\,r^2}\ . $$

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

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

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

Due in class, 4 points. \bigskip

\item{(19a)} 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 in 
class, 4~points.

\item{(19b)} Landau-Lifshitz \textsection 21, Problem~1.
Due in class, 2~points.

\item{(20)} Use the inertial frame of \textsection 15 of the book and
plot the orbits for the four intial conditions of the table in set~5
(best in one figure). Indicate the initial positions. Due February~18
before class, 12 points. 

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Read Landau-Lifshitz \textsection 21. Due February 14.

Read Landau-Lifshitz \textsection 22. Due February 16.


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