Stiff dynamics of electromagnetic two-body motion

Abstract

We study the stability of circular orbits of the electromagnetic two-body problem in an electromagnetic setting that includes retarded and advanced interactions. We give a method to derive the equations of tangent dynamics about circular orbits up to nonlinear terms and we derive the linearized equations explicitly. In particular we study the normal modes of the linearized dynamics that have an arbitrarily large imaginary eigenvalue. These large imaginary eigenvalues define fast frequencies that introduce a fast (stiff) timescale into the dynamics. As an application of Dirac's electrodynamics of point charges with retarded-only interactions, we study the conditions for the two charges to perform a fast gyrating motion of small radius about a circular orbit. The fast gyration defines an angular momentum of the order of the orbital angular momentum, a vector that rotates in the orbital plane at a frequency of the order of the orbital frequency and causes a gyroscopic torque. We explore a consequence of this multiscale solution, i.e; the resonance condition that the angular momentum of the stiff spinning should rotate exactly at the orbital frequency. The resonant orbits turn out to have angular momenta that are integer multiples of Planck's constant to a good approximation. Among the many qualitative agreements with quantum electrodynamics (QED), the orbital frequency of the resonant orbits are given by a difference of two eigenvalues of a linear operator and the emission lines of QED agree with our predictions within a few percent.