Unifying the Hyperbolic and Spherical $2$-Body Problem with Biquaternions

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 and discuss their stability for a strictly attractive potential.