Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 102, 17 pages      arXiv:1407.7919      https://doi.org/10.3842/SIGMA.2014.102

Particle Motion in Monopoles and Geodesics on Cones

Maxence Mayrand
Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec, Canada, H3A 0B9

Received July 31, 2014, in final form November 01, 2014; Published online November 04, 2014

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.

Key words: particle motion; monopoles; geodesics; cones.

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