Particle Motion in Monopoles and Geodesics on Cones

The equations of motion of a charged particle in the field of Yang’s $\mathrm{S}\mathrm{U}(2)$ 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.