Numerical Methods for the 3-dimensional 2-body Problem in the Action-at-a-Distance Electrodynamics

TL;DR

This paper introduces two numerical methods for solving the complex differential equations governing the motion of two charges in 3D action-at-a-distance electrodynamics, addressing energy limitations and convergence issues.

Contribution

It presents a novel numerical approach using Stürmer's extrapolation and an improved iterative scheme for high-energy scenarios in the 3D two-body electrodynamics problem.

Findings

The first method is suitable for shallow energies.

The second method achieves better convergence at high energies.

Both methods extend computational capabilities in action-at-a-distance electrodynamics.

Abstract

We develop two numerical methods to solve the differential equations with deviating arguments for the motion of two charges in the action-at-a-distance electrodynamics. Our first method uses St\"urmer's extrapolation formula and assumes that a step of integration can be taken as a step of light ladder, which limits its use to shallow energies. The second method is an improvement of pre-existing iterative schemes, designed for stronger convergence and can be used at high-energies.