Synge's dynamic problem for two isolated point charges. A new method to find global solutions for Functional Differential Equations System

TL;DR

This paper introduces a novel recursive numerical method to find global solutions for the functional differential equations describing the dynamics of two isolated point charges, addressing Synge's classical problem.

Contribution

A new recursive algorithm is developed to solve the inter-temporal constrained FDEs in Synge's problem, enabling the computation of global solutions with initial conditions.

Findings

Successfully computed quasi-circular solutions for Synge's problem

Demonstrated the effectiveness of the recursive numerical method

Provided insights into the dynamics of isolated charge systems

Abstract

Synge's problem consists in to determine the dynamics of two point electrical charges interacting through their electromagnetic fields, without to take into account the radiation terms due to the self-forces in each charge. We discuss how this problem is related to the question on to establish initial conditions for the electromagnetic fields that are compatible with the two point charges system isolation, that is, the charges are free from the action of external forces. This problem stems from the existence of inter-temporal constraints for the charges trajectories, which implies that the relativistic Newton equations for the charges is not a system of ODEs, but rather a system of Functional Differential Equations (FDEs). We developed a new method to obtain global solutions that satisfies this system of FDEs and a given initial condition for the charges positions and velocities. This method allows the construction of a recursive numerical algorithm that only use integration methods for ODEs systems. Finally, we apply this algorithm to obtain numerical approximations for the quasi-circular solutions that are predicted in Synge's problem.