\documentclass[12pt]{article} % BB Feb 16 2010.
\usepackage{epsfig}
\textheight 9.0in \textwidth  6.0in \topmargin -0.4in \headsep 0.4in  
\oddsidemargin  0.25in  \evensidemargin 0.25in  

\def\sqr#1#2{{\vcenter{\hrule height.#2pt
   \hbox{\vrule width.#2pt height#1pt \kern#1pt
   \vrule width.#2pt} \hrule height.#2pt}}}
\def\square{{\mathchoice\sqr64\sqr64\sqr{4.2}3\sqr{3.0}3}}
\def\undertext#1{$\underline{\smash{\hbox{#1}}}$}
\pagestyle{empty}

\begin{document}

\begin{center}
{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 8}\\%
\vspace {3mm} (February 19, 2010) \\
\end{center}
\setcounter{equation}{0}

\begin{description}

\item{(27a)} Kepler problem: Use unit with $G=1$ for the gravitational
constant and the initial conditions given in the table below (you can 
download the data in the file {\tt Korbits.txt}) to plot the four 
orbits in the coordinate system used in Eq.~(39) of the Notes.
% chapter~8 of T\&M ($r_{\min}$ called perihelion or pericenter 
% at $\phi=0$). 
Make first a table of $L,\, E,\, p,\ e$. Then, put all orbits in 
one plot. Hint: For the first case $e=0.138826$.
Due February~24 before class (10~points).

\item{(27b)} Get the initial positions into the plot of the previous 
homework. February~26 before class (4~extra points).

\item{(27c)} Find positions and velocities in the original system
at time $t=1.5$ (arbitrary units and $G=1$). Ten extra points
when turned in by March~15 or earlier.

\end{description}

\begin{table}[th]   % Generated by porbits.f in KeplerExact. Different
\centering \bigskip % numbers (here->there, e ellipse, h hyperbola): 
                    % 1->14 e, 2->8 h, 3->2 e, 4->12 h.
\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline
 \multicolumn{1}{|c|}{ } & \multicolumn{2}{c|}{Masses}  & 
 \multicolumn{3}{c|}{Initial Positions}  &
 \multicolumn{3}{c|}{Initial Velocities}              \\ \hline
\#&$i$&$m_i$&$x^1_{i,0}$     &$x^2_{i,0}$       &$x^3_{i,0}$
&$\dot{x}^1_{i,0}$&$\dot{x}^2_{i,0}$&$\dot{x}^3_{i,0}$\\ \hline
1&$1$&0.651& 0.585&-0.238&-0.755&-0.828&-0.865&-0.726\\ \hline
 &$2$&0.931&-0.096& 0.000& 0.357&-0.209& 0.107&-0.660\\ \hline
2&$1$&1.510& 0.460&-0.359&-0.234&-0.918&-0.941&-0.323\\ \hline
 &$2$&0.126&-0.066&-0.090&-0.809& 0.789& 0.788& 0.620\\ \hline
3&$1$&1.328&-0.125& 0.898& 0.194&-0.452& 0.172& 0.125\\ \hline
 &$2$&1.999&-0.449&-0.085&-0.454&-0.976&-0.990&-0.968\\ \hline
4&$1$&0.180& 0.204&-0.968&-0.753&-0.811&-0.632& 0.784\\ \hline
 &$2$&1.560&-0.889&-0.979& 0.854&-0.323&-0.774&-0.533\\ \hline
\end{tabular}
\caption{Initial conditions for the Kepler problem (arbitrary units 
and $G=1$).} \label{tab_orbits} \end{table} % \vspace*{0.2cm}

\begin{description}

\item{(28)} 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~$e$
($p$ and $e$ are Landau-Lifshitz notation in M\&T it is $\alpha$
and $\epsilon$). Due February~19 in class 4~points.

\item{(29)} Discuss the $\vec{L}=0$ case of the Kepler problem
using $r=|\vec{r}|$ as defined before. Due February~19 in class
4~points.
\begin{enumerate}
\item Are there turning points? Sketch the potential together
      with assumed $E_{\rm cm}$ energy values.
\item For which initial conditions and energies $E_{\rm cm}$ will
the objects collide or not collide?
\item Find $r(t)$ for the $E_{\rm cm}=0$ case.
\end{enumerate}

\end{description}

% \begin{figure}[tb] \vspace{5pc}
% \centerline{\hbox{ \psfig{figure=figs/ladder.eps,width=7.5cm} }}
% \caption{Ladder for assignment~14.} \label{fig_lader} \end{figure}
\end{document}
