Constructing the Hyperbolic Plane as the reduction of a three-body problem

TL;DR

This paper demonstrates how to derive the hyperbolic plane and its geodesic flow by reducing a specific three-body problem in Euclidean space using the Jacobi-Maupertuis metric, revealing a deep connection between classical mechanics and hyperbolic geometry.

Contribution

It introduces a novel reduction method that constructs the hyperbolic plane from a three-body problem with a specific potential, linking geometric structures with dynamical systems.

Findings

Hyperbolic plane constructed via three-body problem reduction

Geodesic flow corresponds to the reduced system

Uses Jacobi-Maupertuis metric for the reduction

Abstract

We construct the hyperbolic plane with its geodesic flow as the scale plus symmetry reduction of a three-body problem in the Euclidean plane. The potential is $-I/\Delta^2$ where $I$ is the triangle's moment of inertia and $\Delta$ its area. The reduction method uses the Jacobi-Maupertuis metric, following the author's earlier paper "Putting Hyperbolic Pants on a Three-body Problem".