Particle Motion in Monopoles and Geodesics on Cones

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.