Hidden symmetry of hyperbolic monopole motion

TL;DR

This paper explores the dynamics of hyperbolic monopoles, revealing a hidden symmetry that relates their motion to geodesic paths on a generalized multi-centre Taub-NUT space, with detailed analysis of special cases.

Contribution

It introduces a new geometric framework for hyperbolic monopole motion, generalizing the multi-centre Taub-NUT metric and identifying conserved quantities like the Runge-Lenz vector.

Findings

Hyperbolic monopole motion corresponds to geodesic flow on a submanifold of moduli space.

The metric on this submanifold generalizes the multi-centre Taub-NUT metric.

The one-centre case admits a conserved Runge-Lenz vector.

Abstract

Hyperbolic monopole motion is studied for well separated monopoles. It is shown that the motion of a hyperbolic monopole in the presence of one or more fixed monopoles is equivalent to geodesic motion on a particular submanifold of the full moduli space. The metric on this submanifold is found to be a generalisation of the multi-centre Taub-NUT metric introduced by LeBrun. The one centre case is analysed in detail as a special case of a class of systems admitting a conserved Runge-Lenz vector. The two centre problem is also considered. An integrable classical string motion is exhibited.