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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 12}\\%
\vspace {3mm} (March 12, 2010) \\
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\item{(42)} 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 2 before class (5~points).

\item{(43)} The matrix $L$ is defined by
$$ L\ =\ \left( \matrix{ 
  l^0_{\ 0} & l^0_{\ 1} & l^0_{\ 2} & l^0_{\ 3} \cr
  l^1_{\ 0} & l^1_{\ 1} & l^1_{\ 2} & l^1_{\ 3} \cr
  l^2_{\ 0} & l^2_{\ 1} & l^2_{\ 2} & l^2_{\ 3} \cr
  l^3_{\ 0} & l^3_{\ 1} & l^3_{\ 2} & l^3_{\ 3} \cr} \right) $$
\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 April 7 in class. 
\medskip

\item{(44)} In the following $K_1$ and $S_3$ are generators as 
defined in class.
\begin{enumerate}
\item Calculate $\exp(-\zeta\,K_1)$ and explain its physical
meaning (3 points).
\item Calculate $\exp(-\phi\,S_3)$ and explain its physical
meaning (3 points).
\end{enumerate}
Due April 12 before class. 
\medskip

\item{(45)} Consider the spaceship journey again and plot versus the 
proper time $\tau$ the following quantities (due April~5 before 
class, 10~points):
\begin{enumerate}
\item The time on earth at which the spaceship, seen from earth,
      is at its position at time $\tau$.
\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}
 
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\item 2 points for the first correct guess about the figure next
page. Done (went to Evan).
\end{enumerate}

\item{(45)} Continuation of (43). Seen from earth, how far away did 
the spaceship travel? Express the result in light years. 
Due March~20 before class (6~points). 
 
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