Stiff spinning torus of electromagnetic two-body motion

Abstract

We study an orbit of the electromagnetic two-body problem that involves a fast (stiff) spinning motion about a circular orbit. We give a multiscale method of solution that solves for the fast timescale first. The solvability condition of the asymptotic expansion demands resonances for the fast dynamics. The stiff resonant tori are found precisely in the atomic magnitude and agree with many features of the Bohr atom in quantitative and qualitative detail; We calculate every first emission line of the first 13 observable spectroscopic series of hydrogen within a few percent deviation. The resonant orbits have angular momenta that are approximate multiples of Planck's constant and the emitted frequencies are given by a difference of two linear eigenvalues. This Lorentz-invariant two-body dynamics exhibts the phenomenon of resonant dissipation, i.e., the metastable dynamics radiates the center-of-mass energy while the particles perform fast spinning oscillations of small amplitude about a circular orbit, a collective radiative recoil.