Stability, Periodic Solution and Kam Tori in the Circular Restricted (N+1)-Body Problem on S3 and H3

In this article, we define a circular restricted (N+1)-body problem on the surfacesM(kappa)(3), with. = +/- 1. The motion of the primaries corresponds to an elliptic relative equilibria studied in Diacu (Relative equilibria of the curved N-body problem. Atlantis Studies in Dynamical Systems, Atlantis Press, Paris, 2012), where N identical mass particles are rotating uniformly at the vertices of a regular polygon placed at a fixed parallel of a maximal sphere. By introducing rotating coordinates, this problem gives rise to a 3 d.o.f. Hamiltonian system. This problem has an equilibrium point placed at the poles of S-3 and the vertex of H-3, for any value of the parameters. We give information about the linear and nonlinear stability of these equilibria. Finally, we carry out a study about the existence of periodic solutions and KAM tori.