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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 11}\\%
\vspace {3mm} (March 23, 2009) \\
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\item{(47)} The matrix $L$ is defined by
$$ L\ =\ \left( \matrix{ 
  l^0_{\ 0} &amp; l^0_{\ 1} &amp; l^0_{\ 2} &amp; l^0_{\ 3} \cr
  l^1_{\ 0} &amp; l^1_{\ 1} &amp; l^1_{\ 2} &amp; l^1_{\ 3} \cr
  l^2_{\ 0} &amp; l^2_{\ 1} &amp; l^2_{\ 2} &amp; l^2_{\ 3} \cr
  l^3_{\ 0} &amp; l^3_{\ 1} &amp; l^3_{\ 2} &amp; l^3_{\ 3} \cr} \right) $$
\begin{enumerate}
\item Calculate $-g\, L$ (2 points).
\item Write down the transpose matrix $\tilde{L}$ (2 points).
\item Calculate $\tilde{L}\,g$ (2 points).
\item Compare 1.\ and 2.\ to find the general form
      of $L$ ({\it i.e.} use $\tilde{L}\, g = -g\, L$,
      4 points).
\item Obtain the same result by discussing the elements of the
      equation $g^{\alpha\beta} \tilde{l}_{\beta}^{\ \gamma} 
      g_{\gamma\delta} = - l^{\alpha}_{\ \delta}$ (4 extra 
      points. Hint: Do the contractions first.)
\end{enumerate}
Due March 25 in class. 
\medskip

\item{(48)} In the following $K_1$ and $S_3$ are generators as 
defined in class.
\begin{enumerate}
\item Write down the Taylor expansions of the functions $\cosh(x)$, 
      $\cos(x)$, $\sinh(x)$ and $\sin(x)$ (4 points).
\item Calculate $\exp(-\zeta\,K_1)$ and explain its physical
meaning (3 points).
\item Calculate $\exp(-\omega\,S_3)$ and explain its physical
meaning (3 points).
\end{enumerate}
Due March 27 in class. 
\medskip

\item{(49)} Consider the spaceship journey a last time and plot
versus the proper time the following quantities (due March 30 
before class, 10~points):
\begin{enumerate}
\item The time on earth at which the spaceship, seen from earth,
      is at the particular position.
\item The distance from earth as seen from earth.
\item The date of news received from earth, which is transmitted
      at the speed of light.
\end{enumerate}

\item{(50)} Derive the Taylor expansions of the functions 
$\exp(x)$, $\cosh(x)$, $\cos(x)$, $\sinh(x)$ and $\sin(x)$ about 
$x=0$ from the equation
$$ f(x) =\sum_{n=0}^{\infty} f^{(n)}(0)\,\frac{x^n}{n!} $$
by calculating $f^{(n)}(0)$ for each case explicitly. Do NOT
use Euler's formula. Due April 1 before class (5~points).
 
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