Lagrange points of the restricted three-body problem in spaces of constant curvature

Abstract

We continue the study initiated by Kilin ({\em Reg. Chaot. Dyn.} 4, (1999)) and by Martínez and Simó ({\em Celest. Mech. Dynam. Astronom.} 128, (2017)) on the classification and stability of the relative equilibria of the restricted three-body problem in two-dimensional spaces of constant curvature, which generalize the classical Lagrange points of the planar problem. After formulating the problem as an autonomous Lagrangian system with two degrees of freedom, whose only parameters are the curvature $κ$ and the mass ratio $μ$ of the primaries, we establish several classification results for the case $κ>0$ by combining analytical methods with computer-assisted proofs. These results provide rigorous confirmation of phenomena for which previously only numerical evidence was available. We also provide topological explanations for the qualitative differences between the behavior of relative equilibria in positive curvature and that observed in the planar and negative-curvature cases. Our analysis focuses on the regime of small $μ$ and indicates that positive curvature has a stabilizing effect on the triangular equilibria $\Ll_4$ and $\Ll_5$, whereas negative curvature appears to have the opposite effect.