The n-body problem in spaces of constant curvature. Part I: Relative equilibria

Abstract

We extend the Newtonian n-body problem of celestial mechanics to spaces of curvature κ = constant and provide a unified framework for studying themotion. In the 2-dimensional case, we prove the existence of several classes of relative equilibria, including the Lagrangian and Eulerian solutions for any κ = 0 and the hyperbolic rotations for κ <0. These results lead to a new way of understanding the geometry of the physical space. In the end we prove Saari's conjecture when the bodies are on a geodesic that rotates elliptically or hyperbolically. © Springer Science+Business Media, LLC 2012.