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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 6}\\%
\vspace {3mm} (February 8, 2010) \\
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\item{(22a)} Consider the Lagrangian
$$ L\ =\ \frac{1}{2}\,m\,\dot{x}^2 - \frac{1}{2}\,k\,x^2\ .$$
\begin{enumerate}
\item Name the system which this Lagrangian describes.
\item Is the energy conserved? Y/N.
\item Is the (linear) momentum conserved? Y/N.
\end{enumerate}
Due February 8 in class (3 extra points).

\item{(22b)} 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) (2 points)?
  \item Define the center of mass vector $\vec{R}=\vec{R}(t)$ by 
        $M\dot{\vec{R}}=m_1\dot{\vec{r}}_1+m_2\dot{\vec{r}}_2$, where 
        $M=m_1+m_2$ is the total mass, and find the general solution 
        for $\vec{R}(t)$ (2~points).
  \item Use the special solution $\vec{R}=(m_1\vec{r}_1+m_2\vec{r}_2)/M$, 
        to express $\vec{r}_1$ and $\vec{r_2}$ through $\vec{R}$ and
        $\vec{r}$ (3 points). \medskip

The center of mass (cm) systems 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 $M$ (3 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 February~10 in class.
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Read M\&T chapter 8.1 to 8.7. Due February 15.

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