The hyperbolic plane, three-body problems, and Mnëv’s universality theorem

Abstract

We show how to construct the hyperbolic plane with its geodesic flow as the reduction of a three-problem whose potential is proportional to I/Δ2 where I is the moment of inertia of this triangle whose vertices are the locations of the three bodies and Δ is its area. The reduction method follows [11]. Reduction by scaling is only possible because the potential is homogeneous of degree −2. In trying to extend the assertion of hyperbolicity to the analogous family of planar N-body problems with three-body interaction potentials we run into Mn¨ev’s astounding universality theorem which implies that the extended assertion is doomed to fail.