Finite element boundary value integration of Wheeler–Feynman electrodynamics

The electromagnetic two-body problem is solved as a boundary value problem associated to an action functional. We show that the functional is Fréchet differentiable and that its conditions for criticality are the mixed-type neutral differential delay equations with state-dependent delay of Wheeler–Feynman electrodynamics. We construct a finite element method that finds C1C1-smooth solutions when suitable past and future positions of the particles are given as boundary data. The numerical trajectories satisfy a variational problem defined in a finite-dimensional Hermite functional space of C1C1 piecewise-polynomials. The numerical variational problem is solved using a combination of Newton’s method intercalated with boundary adjustments to ensure that the velocity of the solution is continuous with the boundary data. We recover the known circular orbits and compute several other novel trajectories of the Wheeler–Feynman electrodynamics. We also discuss the local convexity of the functional close to the new found trajectories and the possibility of solutions with less regularity.