Variational principle for the Wheeler-Feynman electrodynamics

TL;DR

This paper develops a variational principle for the electromagnetic two-body problem using a Fokker action, analyzing the stability of circular orbits and exploring bifurcations and solution existence.

Contribution

It introduces a Poincare-invariant variational framework for Wheeler-Feynman electrodynamics with boundary conditions, and analyzes the second variation and stability of circular orbits.

Findings

Functional has a local minimum at large-radius circular orbits

Bifurcation occurs at a critical radius where orbits become saddle points

Existence of solutions with perturbed circular boundary conditions

Abstract

We adapt the formally-defined Fokker action into a variational principle for the electromagnetic two-body problem. We introduce properly defined boundary conditions to construct a Poincare-invariant-action-functional of a finite orbital segment into the reals. The boundary conditions for the variational principle are an endpoint along each trajectory plus the respective segment of trajectory for the other particle inside the lightcone of each endpoint. We show that the conditions for an extremum of our functional are the mixed-type-neutral-equations with implicit state-dependent-delay of the electromagnetic-two-body problem. We put the functional on a natural Banach space and show that the functional is Frechet-differentiable. We develop a method to calculate the second variation for C2 orbital perturbations in general and in particular about circular orbits of large enough radii. We prove that our functional has a local minimum at circular orbits of large enough radii, at variance with the limiting Kepler action that has a minimum at circular orbits of arbitrary radii. Our results suggest a bifurcation at some radius below which the circular orbits become saddle-point extrema. We give a precise definition for the distributional-like integrals of the Fokker action and discuss a generalization to a Sobolev space of trajectories where the equations of motion are satisfied almost everywhere. Last, we discuss the existence of solutions for the state-dependent delay equations with slightly perturbated arcs of circle as the boundary conditions and the possibility of nontrivial solenoidal orbits.