Unifying the Hyperbolic and Spherical 2-Body Problem with Biquaternions

TL;DR

This paper unifies the 2-body problem on spherical and hyperbolic spaces using biquaternions, enabling the transfer of results between these geometries and providing a complete classification of relative equilibria in hyperbolic space.

Contribution

It introduces a biquaternionic framework that unifies the 2-body problem on spherical and hyperbolic spaces, facilitating new classifications and insights.

Findings

Complete classification of relative equilibria in hyperbolic 3-space

Unified treatment of spherical and hyperbolic 2-body problems via biquaternions

Results for spherical systems are extended to hyperbolic systems

Abstract

The 2-body problem on the sphere and hyperbolic space are both real forms of holomorphic Hamiltonian systems defined on the complex sphere. This admits a natural description in terms of biquaternions and allows us to address questions concerning the hyperbolic system by complexifying it and treating it as the complexification of a spherical system. In this way, results for the 2-body problem on the sphere are readily translated to the hyperbolic case. For instance, we implement this idea to completely classify the relative equilibria for the 2-body problem on hyperbolic 3-space for a strictly attractive potential.