Variational principle for the Wheeler-Feynman electrodynamics - Mathematical PhysicsReport as inadecuate

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Abstract: We adapt the formally-defined Fokker action into a variational principle forthe electromagnetic two-body problem. We introduce properly defined boundaryconditions to construct a Poincare-invariant-action-functional of a finiteorbital segment into the reals. The boundary conditions for the variationalprinciple are an endpoint along each trajectory plus the respective segment oftrajectory for the other particle inside the lightcone of each endpoint. Weshow that the conditions for an extremum of our functional are themixed-type-neutral-equations with implicit state-dependent-delay of theelectromagnetic-two-body problem. We put the functional on a natural Banachspace and show that the functional is Frechet-differentiable. We develop amethod to calculate the second variation for C2 orbital perturbations ingeneral and in particular about circular orbits of large enough radii. We provethat our functional has a local minimum at circular orbits of large enoughradii, at variance with the limiting Kepler action that has a minimum atcircular orbits of arbitrary radii. Our results suggest a bifurcation at someradius below which the circular orbits become saddle-point extrema. We give aprecise definition for the distributional-like integrals of the Fokker actionand discuss a generalization to a Sobolev space of trajectories where theequations of motion are satisfied almost everywhere. Last, we discuss theexistence of solutions for the state-dependent delay equations with slightlyperturbated arcs of circle as the boundary conditions and the possibility ofnontrivial solenoidal orbits.

Author: Jayme De Luca


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