Particle Motion in Monopoles and Geodesics on Cones

Abstract

The equations of motion of a charged particle in the field of Yang's SU(2) monopole in 5-dimensional Euclidean space are derived by applying the Kaluza-Klein formalism to the principal bundle R⁸∖{0}→R⁵∖{0} obtained by radially extending the Hopf fibration S⁷→S⁴, and solved by elementary methods. The main result is that for every particle trajectory r:I→R⁵∖{0}, there is a 4-dimensional cone with vertex at the origin on which r is a geodesic. We give an explicit expression of the cone for any initial conditions.