Stability, periodic solution and KAM tori in the circular restricted \((N+1)\)-body problem on \({\mathbb{S}}^3\) and \({\mathbb{H}}^3\) (Q6115829)

This paper discusses a restricted (\(N+1\))-body problem in spaces with constant Gaussian curvature \(\kappa\), namely on surfaces \({\mathbb S}^3\) and \({\mathbb H}_\kappa^3\) with \(\sigma = \text{sign} (\kappa)\). By introducing rotating coordinates, this problem gives rise to a Hamiltonian system with three degrees of freedom. The authors are interested in giving information about the linear and nonlinear stability of these equilibria. Also, they carry out a study about the existence of periodic solutions and KAM tori. In the curved \(N\)-body problem the relative equilibria are defined as a particular solution where the mutual distances among the particles remain constant during the motion, i.e., the particle system behaves as a rigid body. Hence, the motion of the primaries in the restricted (\(N+1\))-body problem in spaces with constant Gaussian curvature corresponds to an elliptic relative equilibria studied by \textit{F. Diacu} [Relative equilibria of the curved \(N\)-body problem. Amsterdam: Atlantis Press (2012; Zbl 1261.70001)], where \(N\) equal masses are rotating uniformly at the vertices of a regular polygon placed at a fixed parallel of a maximal sphere. The authors consider elliptic relative equilibria originated by the elliptic rotation \(\kappa\)-positive of \({\mathbb S}_\kappa^3\) and \(\kappa\)-negative of \({\mathbb H}_\kappa^3\), respectively. This configuration is referred to as a \textit{ring configuration}. The problem has an equilibrium point placed at the poles of \({\mathbb S}^3\) and the vertex of \({\mathbb H}^3\), for any value of the parameters. The authors give information about the linear and nonlinear stability of these equilibria. The existence of certain periodic orbits is proved through averaging theory for Hamiltonian systems as well as use of symmetries. The authors study the existence of a family of KAM tori close to an integrable Hamiltonian by means of Han, Li and Yi's theorem [\textit{Y. Han} et al., Ann. Henri Poincaré 10, No. 8, 1419--1436 (2010; Zbl 1238.37018)]. The authors also use numerical computations to find some periodic orbits for the massless particle. Some nice illustrations are provided.