The n-body problem in spaces of constant curvature

TL;DR

This paper extends the classical n-body problem to spaces with constant curvature, revealing new singularities, equilibria, and confirming Saari's conjecture in specific geometric settings.

Contribution

It generalizes the n-body problem to curved spaces, identifies new types of singularities and equilibria, and proves Saari's conjecture for bodies on rotating geodesics.

Findings

Discovery of non-collision singularities at antipodal points for k>0

Existence of hyperbolic relative equilibria for k<0

Proof of Saari's conjecture for bodies on rotating geodesics

Abstract

We generalize the Newtonian n-body problem to spaces of curvature k=constant, and study the motion in the 2-dimensional case. For k>0, the equations of motion encounter non-collision singularities, which occur when two bodies are antipodal. This phenomenon leads, on one hand, to hybrid solution singularities for as few as 3 bodies, whose corresponding orbits end up in a collision-antipodal configuration in finite time; on the other hand, it produces non-singularity collisions, characterized by finite velocities and forces at the collision instant. We also point out the existence of several classes of relative equilibria, including the hyperbolic rotations for k<0. In the end, we prove Saari's conjecture when the bodies are on a geodesic that rotates elliptically or hyperbolically. We also emphasize that fixed points are specific to the case k>0, hyperbolic relative equilibria to k<0, and Lagrangian orbits of arbitrary masses to k=0--results that provide new criteria towards understanding the large-scale geometry of the physical space.