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\begin{document}

%Thirteenth International Conference Ejemplo--Pau on 
%Mathematics and its Applications (Jaca - September 2014).

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                %\begin{Large}   Universidad \\
                %                de Ejemplo\end{Large}
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                        \veryHuge \textsf{\textbf{Periodical path around bodies}}\\[.6ex]
                        \LARGE {\color{rojo}\bfseries Por Ejemplo}%\footnotemark[1]\footnotemark[2]
            \\[0.8ex]
                        \Large
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                        \emph{Grupo de investigacion, Department of Ejemplo\\
                        Universidad de Ejemplo, Spain.}\\
                        \emph{ejemplo@unizar.es}

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\section*{Introduction}

The most common method to compute the ...

To calculate series...

In this work we are interested in the study of geometric shapes.

We calculate the gravitational potential of square orbit.

\section*{The potential of a plate}

First works on the potential of a .

We use the potential obtained by ...

The potential function of a plate  is:
\begin{eqnarray*}
U = G \sigma \sum_{i=1}^{n} (C_{i,{\rm mod}(i+1,n)}L_{i,{\rm
mod}(i+1,n)}/r_{i,{\rm mod}(i+1,n)})
\end{eqnarray*}

\noindent where ${\rm mod}(i+1,n)$ is the modulus n of i+1.

The values inside the sum are defined as follows:

\begin{itemize}
   \item $r_{ij}$ are the sides of the triangle,
   \item $C_{ij}$ are the cross products of the position vectors of the vertices,
   \item $d_{ij}$ are the distances of the point mass to the
   vertices, and
   \item $L_{ij}$ are the logarithms defined by:
\end{itemize}
\begin{eqnarray*}
L_{ij} = \log [(d_i+d_j+r_{ij})/(d_i+d_j-r_{ij})].
\end{eqnarray*}

The equations of motion of a point mass around this plate are:

\begin{eqnarray*}
\ddot x &=& +G \sigma \sum_{i=1}^{n}(y_i-y_{{\rm mod}(i+1,n)}) L_{i,{\rm mod}(i+1,n)}\\
\ddot y &=& -G \sigma \sum_{i=1}^{n}(x_i-x_{{\rm mod}(i+1,n)})
L_{i,{\rm mod}(i+1,n)}
\end{eqnarray*}

It should be noted that ...

  \begin{center}
    \includegraphics[width=0.155\textwidth]{triangulocontour.eps}
    \includegraphics[width=0.15\textwidth]{cuadradocountour2.eps}
  \end{center}

We consider the following plates:

\begin{itemize}
   \item An equilateral triangle with the
origin at the center of mass and with one of its vertices in the
minus x axis.
   \item A square with its sides parallel to the axes.
\end{itemize}

In the figure above we have drawn the countour plots of these
potentials.

\section*{Periodic Orbits for the ...}

We have calculated several Poincar\'e surfaces of section ...

\begin{center}
\includegraphics[width=0.25\textwidth]{ssp-triangulo.eps}
\end{center}

In this surface of section we have both direct and retrograde
orbits. It is possible to ...

\begin{center}
\includegraphics[width=0.15\textwidth]{orbita1.eps}
\includegraphics[width=0.15\textwidth]{orbita2.eps}
\end{center}

The stability i ...

\begin{center}
\includegraphics[width=0.15\textwidth]{zoom-trace-bif.eps}
\includegraphics[width=0.15\textwidth]{energy.eps}
\end{center}

\section*{Future work}

We plan to study\cite{label5} the orbits in the planar faces.

\section*{Acknowledgements}

%We thank Dr. A. Elipe for suggesting to study the triangular plate
%and Dr. R. Barrio for his help on .

This work has been partially financed by the Ministerio de
Educaci\'on, Ciencia y Tecnolog\'{\i}a (Proyecto \#ESP2005-.....).

\begin{thebibliography}{9}

  % Pattern for book entries.
  \bibitem{label1} \textsc{O.D. Kellogg} \emph{Foundations of potential
  theory}, Dover, N.Y, 1929.

  \bibitem{label2} \textsc{W.D. MacMillan}. \emph{The theory of the potential}.
  Dover, N.Y., 1930.

  % Pattern for proceedings. Write the editors' names like
  % authors' names
  \bibitem{label3} \textsc{R. Broucke}. Closed Form Expressions for Some Gravitational Potentials:
  Triangle, Rectangle, Pyramid and Polyhedron.
  In \emph{The Dynamics of Small Bodies in the Solar System, A Major Key to Solar System Studies},
  B. A. Steves and A. E. Roy (eds.), pp. 321. Kluwer Academic Publishers, 1999.

%  \bibitem{label2} \textsc{R. Broucke}. Closed Form Expressions for Some Gravitational Potentials: Triangle,
%  Rectangle, Pyramid and Polyhedron.
%  \emph{AAS/AIAA Spaceflight Mechanics Meeting} \textbf{89}, pp. 1293, 1995.

 % Pattern for articles
  \bibitem{label4} \textsc{B. Werner}. The gravitational potential of a homogeneous polyhedron or don't cut corners.
  \emph{Celestial Mechanics and Dynamical Astronomy} \textbf{59}, pp. 253--278, 1994.

  \bibitem{label5} \textsc{P. Ejemplo}. Periodical orbits around symmetrical planar bodies.
  \emph{In preparation}.

  \bibitem{label6} \textsc{F.J. Mu\~noz-Almaraz, E. Freire, J. Gal\'an, E. Doedel, A. Vanderbauwhede}.
  Continuation of periodic orbits in conservative and Hamiltonian systems.
  \emph{Physica D}, \textbf{181}, pp. 1--38, 2003.

  \bibitem{label7} \textsc{R. Marrio, P. Ejemplo, M. Jara}. VSVO formulation of the Taylor method for the numerical
  solution of ODEs. \emph{Computers and Mathematics with applications}, \textbf{50}, pp. 93--111, 2005.
\end{thebibliography}

\end{multicols}
\end{document}
 