\documentclass[style=aggie]{powerdot}
\title{Newtonian N-Body Problem}
\author{Predrag Puno\v{s}evac\\
         Department of Mathematics\\
         The University of Arizona }
\date{02/13/2007}
\begin{document}
\maketitle
\begin{slide}{Introduction}
   \begin{itemize} 
     \item  How do planets move?
          \begin{enumerate}
           \pause 
        \item {\color{green} According to Aristotle's 
point of view (adopted by Christian church)
                   the Earth is the center of universe.}
           \pause \item {\color{blue}Galileo argued 
           that the Sun is the center of solar system.}
           \end{enumerate}
     \pause \item What is Newton's law of gravitation?

               \pause {\color{red}$F=G\frac{m_1m_2}{r^2}$ }   
     \pause \item What is N-body problem? 

     \pause  { 
The N-body problem is the mathematical study of how heavily bodies move in settings
where dynamics are dictated by Newton's law of gravitation.}
     \pause 
\item {
If the Aristotle was right, how would Newton's low of motion look like?}        
   \end{itemize}
\end{slide}
\begin{slide}{Real Motions of Planets}
    \begin{itemize}
       \item In reality we know that planets revolve around Sun.
          \vskip 0.5cm
       \item We focus on Earth and Mars.    
          \vskip 0.5cm
    \pause   \item What is the shape of their orbits?

    \pause  Their orbits are nearly circular. If the radious of Earth's orbit was 
             2 miles the radious of Mars' orbit would be 3 aproximately 3 miles.
          \vskip 0.5cm
    \pause  \item What are the rotational periods?

\pause It takes one year for Earth to orbit the Sun. It takes about two years for Mars to 
orbit the Sun. 
    \end{itemize}
\end{slide}
\begin{slide}{Trigonometry 101}
       \begin{itemize}
         \item What is the equation of the circle with the center in the origin?

         \item How does the equation of the circle looks like in the trigonometric notation?
         
         \item What about the difference in periods?
       \end{itemize}
\end{slide}
\begin{slide}{Relative Motion of Mars}
      \begin{itemize}
          \item We know that Mars revolves around Sun but 
                   how does the motion of Mars look like to an earthly creature?
      \end{itemize}
\begin{eqnarray*}
\pause r=&r_M-r_E\cr
 =&(3\cos(\pi t),3 \sin (\pi t))-(2\cos(2\pi t),2\sin(2\pi t))\cr
\pause =&\left(3\cos(\pi t)-2(\cos^2(\pi t)-1),3\sin(\pi t)-2\sin(\pi t)\cos(\pi t)\right)\cr
\pause =&(2,0)+\rho(\cos(\pi t),\sin(\pi t)) 
\end{eqnarray*}

where $\rho=3-4\cos(\pi t)$.
\begin{itemize}
          \item What is the graph of the above function?
      \end{itemize}

\pause {\color{red} \it limacon}
\end{slide}
\begin{slide}{Conclusion}
If the Earth was in the centre of universe the Universal Law of Gravitation would be 
attractive-repulsive.
That would be a major inconviniance.
\end{slide}
\end{document}
\endinput

