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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 7}\\%
\vspace {3mm} (February 12, 2010) \\
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\begin{description}

\item{(23)} Assume a central potential and $\vec{L}\ne 0$. 
\begin{enumerate}
  \item Use initial conditions in the CM system, $\vec{r}_0$ and
  $\vec{v}_0$ to define unit vectors $\hat{e}_1$, $\hat{e}_2$ 
  and $\hat{e}_3$ of a Cartesian coordinates system in which
  $$L=|\vec{L}|=\mu\,r^2\,\dot{\phi}\,,~~~\phi_0=0\,,~\dot{\phi}_0>0$$
  holds ($\phi$ azimuth angle).  Due February 12 in class (4 points).

  \item Express the total energy $E_{\rm cm}$ in terms of $r$ and
  $\phi$ and their derivatives (i.e., get rid of $\vec{v}$). Substitute 
  the conserved angular momentum $L$ into your expression. Combine all 
  terms, which contain $r$ but not $\dot{r}$, into the definition of 
  an effective potential $U_{\rm eff}(r)$. Due February 12 in class 
  (4~points). 
\end{enumerate}

\item{(24)} Use (in arbitrary units) the parameters $\alpha=\mu=L=1$ to 
plot the potential and the effective potential of the Kepler problem
$$ U = - \frac{\alpha}{r}~~{\rm and}~~U_{\rm eff}(r) 
     = U(r) + \frac{L^2}{2\,\mu\,r^2} $$
in the range $0<r\le 10$ with the ordinate restricted to the interval
$[-1,0.6]$. Describe the motion for $E_{\rm cm}=U_{\rm eff}^{\min}$,
where $U_{\rm eff}^{\min}=U_{\rm eff}(r^{\min})$ is the minimum of 
the effective potential. Calculate $r^{\min}$ as function of $L$,
$\mu$ and $\alpha$ and give then the numerical value. 
Due February~15 before class (8~points).

\item{(25a)} Separate variables in the equation 
$$ E_{\rm cm} = \frac{\mu}{2}\dot{r}^2 + U_{\rm eff}(r) $$
to derive an integral equation for $t(r)$. Due February~15 in class
(4~points).

\item{(25b)} Use angular momentum conservation and the separation of
variables in the previous problem to derive an integral equation for 
$\phi(r)$. Due February~15 in class (4~points).

\end{description}

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