The n-body problem in spaces of constant curvature

Abstract: We generalize the Newtonian n-body problem to spaces of curvature k=constant, and study the motion in 2 dimensions. We prove the existence of several classes of relative equilibria, including the hyperbolic rotations for k<0. We also classify all homographic solutions of the 3-body case with equal masses. 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.