On the restricted (N + 1)-body problem on surfaces of constant curvature

Abstract

In this paper, we consider a restricted (N+1)-body problem on surfaces Mκ2, where the constant κ≠0 is the Gaussian curvature, which by means of a rescaling can be reduced to κ=±1. This problem consists in the study of the dynamics of an infinitesimal mass particle attracted by N primaries of identical masses describing elliptic relative equilibria of the N-body problem on Mκ2, i.e., a solution where the primaries are rotating uniformly with constant angular velocity ω on a fixed parallel of S2 or H2 and placed at the vertices of a regular polygon. In a rotating frame, this problem yields a two degrees of freedom Hamiltonian system. The goal of this paper is to describe analytically some dynamics features for κ=±1. Precisely, we study the relative location of equilibria, obtaining, in particular, that the poles of S2 and vertex of H2 represent equilibrium points for any value of the parameters. Thus, analysis of the nonlinear stability of these equilibria is carried out, as well as two types of bifurcations are detected: Hamiltonian-Hopf and N-bifurcation. Additionally, we prove the existence of a family of Hill's orbits and a family of periodic orbits when the primaries are located near the poles of S2 or the vertex of H2. Finally we prove the existence of KAM 2-tori related to these periodic orbits.