Geodesics of charged particle in electromagnetic field

TL;DR

This paper explores the motion of charged particles in electromagnetic fields using Finsler geometry, deriving metrics and geodesics influenced by external magnetic fields and surface slopes.

Contribution

It introduces geometric solutions for charged particle trajectories in electromagnetic fields within Finsler geometry, including new metric classes and effects of magnetic fields on geodesics.

Findings

Metrics belong to ({\alpha}, {\beta})-class with and without slopes

Magnetic field introduces an additional parameter in the metric

Geodesics and indicatrices are estimated for specific surfaces

Abstract

We report calculations about the motion of a charged particle in an external electric and magnetic field. The metric for the particle moving on a slope with non-zero traction and coefficient of friction is also evaluated for weak fields. We have geometric solutions in terms of Finsler geometry. We show that our solution metrics belong to the ({\alpha}, \b{eta})-metric class for cases with and without motion on a slope. Further, the external magnetic field is manifested in an additional parameter in the metric. The geodesic spray coefficients under the influence of magnetic field have also been calculated. Finally, we have estimated the indicatrices and geodesics for slippery plane and cone.