Reduction and relative equilibria for the 2-body problem in spaces of constant curvature

TL;DR

This paper classifies and analyzes the stability of relative equilibria in the two-body problem on surfaces of constant curvature, revealing stability conditions and bifurcations in hyperbolic and spherical geometries.

Contribution

It provides a comprehensive classification of relative equilibria and their stability in curved spaces, using global reduction and bifurcation analysis, which was not previously detailed.

Findings

Hyperbolic relative equilibria are unstable.

Elliptic relative equilibria are stable when close, unstable when far.

Unique relative equilibrium exists on the sphere for different masses, with stability depending on the angle.

Abstract

We consider the two-body problem on surfaces of constant non-zero curvature and classify the relative equilibria and their stability. On the hyperbolic plane, for each q>0 we show there are two relative equilibria where the masses are separated by a distance q. One of these is geometrically of elliptic type and the other of hyperbolic type. The hyperbolic ones are always unstable, while the elliptic ones are stable when sufficiently close, but unstable when far apart. On the sphere of positive curvature, if the masses are different, there is a unique relative equilibrium (RE) for every angular separation except {\pi}/2. When the angle is acute, the RE is elliptic, and when it is obtuse the RE can be either elliptic or linearly unstable. We show using a KAM argument that the acute ones are almost always nonlinearly stable. If the masses are equal there are two families of relative equilibria: one where the masses are at equal angles with the axis of rotation (`isosceles RE') and the other when the two masses subtend a right angle at the centre of the sphere. The isosceles RE are elliptic if the angle subtended by the particles is acute and is unstable if it is obtuse. At {\pi}/2, the two families meet and a pitchfork bifurcation takes place. Right-angled RE are elliptic away from the bifurcation point. In each of the two geometric settings, we use a global reduction to eliminate the group of symmetries and analyse the resulting reduced equations which live on a 5-dimensional phase space and possess one Casimir function.