The curved symmetric $ 2 $– and $ 3 $–center problem on constant negative surfaces

Abstract

<p style='text-indent:20px;'>We study the motion of the negative curved symmetric two and three center problem on the Poincaré upper semi plane model for a surface of constant negative curvature <inline-formula><tex-math id="M3">\begin{document}$ \kappa $\end{document}</tex-math></inline-formula>, which without loss of generality we assume <inline-formula><tex-math id="M4">\begin{document}$ \kappa = -1 $\end{document}</tex-math></inline-formula>. Using this model, we first derive the equations of motion for the <inline-formula><tex-math id="M5">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-and <inline-formula><tex-math id="M6">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>-center problems. We prove that for <inline-formula><tex-math id="M7">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>–center problem, there exists a unique equilibrium point and we study the dynamics around it. For the motion restricted to the invariant <inline-formula><tex-math id="M8">\begin{document}$ y $\end{document}</tex-math></inline-formula>–axis, we prove that it is a center, but for the general two center problem it is unstable. For the <inline-formula><tex-math id="M9">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>–center problem, we show the non-existence of equilibrium points. We study two particular integrable cases, first when the motion of the free particle is restricted to the <inline-formula><tex-math id="M10">\begin{document}$ y $\end{document}</tex-math></inline-formula>–axis, and second when all particles are along the same geodesic. We classify the singularities of the problem and introduce a local and a global regularization of all them. We show some numerical simulations for each situation.</p>