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\begin{center}
{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 5}\\%
\vspace {3mm} (February 4, 2011) \\
\end{center}
\setcounter{equation}{0}

\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}
% \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}

\caption{Initial conditions for the Kepler problem (arbitrary units 
and set $G=1$).} \label{tab_orbits} \end{table} % \vspace*{0.2cm}

\begin{description}

\item{(15)} Calculate the angular momentum vectors in the CM frame
for the initial conditions given in the table above (you can download
the data as text file Kepler.txt).
Due February~7 before class 8~points.

\item{(16)} Let the interaction of two point particles be described 
by a potential which depends only on their distance:
$$ {\cal L} = \frac{m_1}{2}\vec{v}_1^{\,2}+\frac{m_2}{2}\vec{v}_2^{\,2}
           - U(r)\,,~~~r=|\vec{r}|\,,\ \vec{r}=\vec{r}_1-\vec{r}_2\ . $$
\begin{enumerate}
  \item Is the energy of this system conserved (with reason) (1 point)?
  \item Define the center of mass vector by $\vec{R}=(m_1\vec{r}_1+
        m_2\vec{r}_2)/M$, $M=m_1+m_2$ and express $\vec{r}_1$ and 
        $\vec{r_2}$ through $\vec{R}$ and $\vec{r}$ (2 points). \medskip

The center of mass (cm) frame is defined by $\vec{R}(t)=0$. Show 
the following equalities in the cm system: \medskip

  \item $T_{\rm cm} = m_1\,\vec{v}_1^{\,2}/2+m_2\,\vec{v}_2^{\,2}/2 
  = \mu\,\vec{v}^{\,2}/2$ with $\vec{v}=\dot{\vec{r}}$ and $\mu$ the 
  reduced mass. Express $\mu$ through $m_1$, $m_2$ and $M$ (2 points).
  \item $\vec{L}=\vec{r}_1\times\vec{p}_1+\vec{r}_2\times\vec{p}_2=
         \mu\,\vec{r}\times\vec{v}$ (2 points).
\end{enumerate}
Due in class.

\item{(17)} Plot the effective potentials corresponding to the initial 
conditions of the table together with the CM energies (the Potential
is $U(r)=-G\,m_1m_2/r$). Due February~11 before class 12~points.

\end{description}
\end{document}
