Particle Motion in Monopoles and Geodesics on Cones

TL;DR

This paper derives equations of motion for a charged particle in a Yang's SU(2) monopole field in 5D space, showing that all trajectories are geodesics on specific 4D cones, with explicit formulas provided.

Contribution

It introduces a novel geometric characterization of particle trajectories as geodesics on cones in 5D space, derived via Kaluza-Klein formalism and explicit solutions.

Findings

Particle trajectories are geodesics on 4D cones.

Explicit cone equations are provided for any initial conditions.

The equations of motion are derived using elementary methods.

Abstract

The equations of motion of a charged particle in the field of Yang's $\mathrm{SU}(2)$ monopole in 5-dimensional Euclidean space are derived by applying the Kaluza-Klein formalism to the principal bundle $\mathbb{R}^8\setminus\{0\}\to\mathbb{R}^5\setminus\{0\}$ obtained by radially extending the Hopf fibration $S^7\to S^4$, and solved by elementary methods. The main result is that for every particle trajectory $\mathbf{r}:I\to\mathbb{R}^5\setminus\{0\}$, there is a 4-dimensional cone with vertex at the origin on which $\mathbf{r}$ is a geodesic. We give an explicit expression of the cone for any initial conditions.