Kepler unbound: some elegant curiosities of classical mechanics

TL;DR

This paper explores two classical mechanics systems with conserved Laplace-Runge-Lenz vectors, revealing elegant variations of Kepler's laws and introducing a new modified Kepler third law for BPS monopoles.

Contribution

It presents a detailed analysis of two exotic systems, highlighting their unique Kepler-like laws and deriving a new modified Kepler third law for BPS monopoles.

Findings

Both systems conserve a Laplace-Runge-Lenz vector.

Each system follows a variation of Kepler's three laws.

A new modified Kepler third law is derived for BPS monopoles.

Abstract

We describe two exotic systems of classical mechanics: the McIntosh-Cisneros-Zwanziger ('MICZ') Kepler system, of motion of a charged particle in the presence of a modified dyon; and Gibbons and Manton's description of the slow motion of well-separated solitonic ('BPS') monopoles using Taub NUT space. Each system is characterized by the conservation of a Laplace-Runge-Lenz vector, and we use elementary vector techniques to show that each obeys a subtly different variation on Kepler's three laws for the Newton/Coulomb two-body problem, including a new modified Kepler third law for BPS monopoles.