A numerical method to calculate geodesics of charged particle in electromagnetic field

We report an algorithm and its numerical implementation using ordinary differential equations derived from Finsler geometry to estimate the geodesics of a charged particle moving on a surface in the presence of external, static electric and magnetic fields. The equations derived here inherently incorporate factors such as traction and surface roughness through two parameters, traction coefficient and coefficient of dynamic friction. This allows us to a prepare a numerical model that avoids the use of additional differential equations and calculations to correct for such affects. We show that the algorithm can reliably and quickly predict the shortest path a charged particle takes on the inner surface of a cone. The results of presented here should find use in practical situations such as plasma physics, ion-beam thrusters, porous electrodes for electrochemical energy storage, ion-channels in cells of animals and plants.