Relative equilibria of the restricted three-body problem in curved spaces

© 2017, Springer Science+Business Media Dordrecht. We use a formulation of the N-body problem in spaces of constant Gaussian curvature, κ∈ R, as widely used by A. Borisov, F. Diacu and their coworkers. We consider the restricted three-body problem in S2 with arbitrary κ> 0 (resp. H2 with arbitrary κ< 0) in a formulation also valid for the case κ= 0. For concreteness when κ> 0 we restrict the study to the case of the three bodies at the upper hemisphere, to be denoted as S+2. The main goal is to obtain the totality of relative equilibria as depending on the parameters κ and the mass ratio μ. Several general results concerning relative equilibria and its stability properties are proved analytically. The study is completed numerically using continuation from the κ= 0 case and from other limit cases. In particular both bifurcations and spectral stability are also studied. The H2 case is similar, in some sense, to the planar one, but in the S+2 case many differences have been found. Some surprising phenomena, like the coexistence of many triangular-like solutions for some values (κ, μ) and many stability changes will be discussed.