  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.
