We show how to construct the hyperbolic plane with its geodesic flow as the reduction of a three-problem whose potential is proportional to $I/\Delta^2$ where $I$ is the moment of inertia of this triangle whose vertices are the locations of the three bodies and $\Delta$ 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 Mnëv’s astounding universality theorem which implies that the extended assertion is doomed to fail.
Keywords:
Jacobi–Maupertuis metric, reduction, Mnev’s Universality Theorem, three-body forces, Hyperbolic metrics
Citation:
Montgomery R., The Hyperbolic Plane, Three-Body Problems, and Mnëv’s Universality Theorem, Regular and Chaotic Dynamics,
2017, Volume 22, Number 6,
pp. 688–699