A new method for finding global solutions to Synge’s electromagnetic problem

Abstract

Synge’s problem consists in determining the dynamics of two point electrical charges interacting through their electromagnetic fields, without taking 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 in 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 ordinary differential equations (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.