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

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 \begin{document}$ \kappa $\end{document} , which without loss of generality we assume \begin{document}$ \kappa = -1 $\end{document} . Using this model, we first derive the equations of motion for the \begin{document}$ 2 $\end{document} -and \begin{document}$ 3 $\end{document} -center problems. We prove that for \begin{document}$ 2 $\end{document} –center problem, there exists a unique equilibrium point and we study the dynamics around it. For the motion restricted to the invariant \begin{document}$ y $\end{document} –axis, we prove that it is a center, but for the general two center problem it is unstable. For the \begin{document}$ 3 $\end{document} –center problem, we show the non-existence of equilibrium points. We study two particular integrable cases, first when the motion of the free particle is restricted to the \begin{document}$ y $\end{document} –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.