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 and discuss their stability for a strictly
attractive potential.