Variational Electrodynamics of Atoms

We generalize Wheeler-Feynman electrodynamics with a variational problem for trajectories that are required to merge continuously into given past and future boundary segments. We prove that the boundary-value problem is well posed for two classes of boundary data. The well-posed solution in general has velocity discontinuities, henceforth a broken extremum. Along regular segments, broken extrema satisfy the Euler-Lagrange neutral differential delay equations with state-dependent deviating arguments. At points where velocities are discontinuous, broken extrema satisfy the Weierstrass-Erdmann conditions that energies and momenta are continuous. Electromagnetic fields of the finite trajectory segments are derived quantities that can be extended to a bounded region B of spacetime. Extrema with a finite number N of velocity discontinuities have extended fields defined in B with the possible exception of N spherical surfaces, and satisfy the integral laws of classical electrodynamics for most surfaces and curves inside B. As an application, we study the hydrogenoid atomic model with mass ratio varying by three orders of magnitude to include hydrogen, muonium and positronium. For each model we construct globally bounded trajectories with vanishing far-fields using periodic perturbations of circular orbits. Our model uses solutions of the neutral differential delay equations along regular segments and a variational approximation for the head-on collisional segments. Each hydrogenoid model predicts a discrete set of finitely measured neighbourhoods of periodic orbits with vanishing far-fields right at the correct atomic magnitude and in quantitative and qualitative agreement with experiment and quantum mechanics. The spacings between consecutive discrete angular momenta agree with Planck’s constant within thirty-percent, while orbital frequencies agree with a corresponding spectroscopic line within a few percent. Received 12 May 2013, Accepted 6 July 2013, Scheduled 18 July 2013 * Corresponding author: Jayme De Luca ([email protected]).