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\title{Rectangular orbits of the curved 4-body problem}
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\markboth{Florin Diacu and Brendan Thorn}{Orbits in the constant curved $N$-body problem (PRELIMINARY NOTES)}
\author{\begin{center}
{\bf Florin Diacu}$^{1,2}$ and {\bf Brendan Thorn}$^2$\\
\smallskip
{\footnotesize $^1$Pacific Institute for the Mathematical Sciences\\
and\\
$^2$Department of Mathematics and Statistics\\
University of Victoria\\
P.O.~Box 3060 STN CSC\\
Victoria, BC, Canada, V8W 3R4\\
diacu@uvic.ca and bthorn@uvic.ca\\
}\end{center}
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\section*{\bf Abstract of the Poster}
We consider the 4-body problem in spaces of constant curvature and study the 2-dimensional (2D) and 3-dimensional (3D) existence of rectangular relative equilibria, i.e.\ configurations that maintain constant mutual distances, and rotopulsators, i.e.\ orbits that both rotate and change size in time, \cite{01}, \cite{02}, \cite{03}. Our results show that certain types of orbits don't exist, but we put into the evidence several large classes of rectangular orbits, \cite{04},
\cite{05}.
\begin{thebibliography}{99}
\bibitem{01} F. Diacu, On the singularities of the curved $N$-body problem, {\it Trans.\ Amer.\ Math.\ Soc.} {\bf 363}, 4 (2011), 2249--2264.
\bibitem{02} F.\ Diacu, Relative equilibria in the 3-dimensional curved $n$-body problem, {\it Memoirs Amer.\ Math.\ Soc.} (to appear).
\bibitem{03} F.\ Diacu, {\it Relative Equilibria of the Curved $N$-Body Problem}, Atlantis Series in Dynamical Systems, vol.\ 1, Atlantis Press, 2012.
\bibitem{04} F.\ Diacu and S.\ Kordlou, {\it Rotopulsators of the curved $N$-body problem}, arXiv:1210.4947 (40 pages).
\bibitem{05} F.~Diacu, R.~Mart\'inez, E.\ P\'erez-Chavela, and C.\ Sim\'o, On the stability of tetrahedral relative equilibria in the positively curved 4-body problem, arXiv:1204.5729,
{\it Physica D} (to appear).
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