% BB 09 04 25
\documentclass[12pt]{article}
\textheight 9.5in \textwidth 6.0in
\topmargin -0.7in \headsep   0.4in  
\oddsidemargin 0.25in \evensidemargin 0.25in  

\def\pmbf#1{{\mbox{\boldmath${#1}$}}}
\def\undertext#1{$\underline{\smash{\hbox{#1}}}$}
\pagestyle{empty}

\begin{document}

%----------------------------------------------------%
%%%%%%%%%%%%%%%%% ADVANCED DYNAMICS %%%%%%%%%%%%%%%%
%----------------------------------------------------%
\begin{center}
{\bf FINAL ADVANCED DYNAMICS -- PHY 4241/5227\ (April 27, 2011)}\\
\end{center}
\setcounter{equation}{0}
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\bf PROBLEM 1 (25 points)}
\medskip \noindent
Assume a Lagrangian $L=L(\{q_i\},\{\dot{q}_i\},t)$ where $q_i,\,i=1,
\dots,n$ are generalized coordinates, $\dot{q}_i,\,i=1,\dots,n$ are 
generalized velocities and $t$ is the time.
\begin{enumerate}
\item Write down the principle of least action.
\item Derive the Euler-Lagrange equations from the principle of least 
      action.
\item Assume that the Lagrangian is invariant under translations
      $q_i\to q_i'=q_i+\epsilon_i$. Find the corresponding conserved
      quantities.
\end{enumerate}
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\bf PROBLEM 2 (25 points)}
\begin{enumerate}
\item Write down the two basic postulates of the Special Theory of 
      Relativity.
\item The lifetime of a pion is $\tau\!=\!2.6\times10^{-8}$~s (with 
      respect to its rest frame). Assume the pion travels in a collider 
      experiment at 99\% of the speed of light with respect to the Lab
      frame. 
      \begin{enumerate}
      \item How long is the observed lifetime of the pion in the Lab 
            frame?
      \item How far does the pion travel in the Lab frame during one
            lifetime?
      \end{enumerate}
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\bf PROBLEM 3 (25 points)}
\medskip \noindent
\noindent The electromagnetic field tensor is given by
$$ (F^{\alpha\beta}) = \left( \matrix{ 
			      \ 0& -E^x& -E^y& -E^z \cr
			      E^x&\ 0  & -B^z&\ B^y \cr
			      E^y&\ B^z&\ 0  & -B^x \cr
			      E^z& -B^y&\ B^x&\ 0 \cr}\right) \ . $$
Find (the Einstein summation convention is assumed for identical
indices):
\begin{enumerate}
\item $F^{\alpha}_{\ \ \alpha}$. 
\item The matrix $(F_{\alpha\beta})$.
\item $F_{\alpha\beta} F^{\alpha\beta}$.
\end{enumerate}
\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%
\centerline{\bf PROBLEM 4 (25 points)}
\medskip \noindent
An electron and a positron, each with mass equal to $0.511~MeV$, 
annihilate at rest into two photons. 
\begin{enumerate}
\item Chose the rest frame of the positronium, such that one of the 
      photons moves in positive $x=x^1$ direction. For each photon 
      find the momentum 4-vector.

\item Find for one of the photons the 4-momentum in a frame that moves 
      at a velocity $c\beta\hat{x}$ along the previously defined $x^1$ 
      direction with respect to the rest frame of the positronium.

\item Suppose that the annihilation took place $10^9$ years ago in a 
      galaxy that is receding from us at $\beta = {4/5}$. 
      What is the energy of the photon that we observe?
\end{enumerate}
\vfill
\end{document}

