Minimizers with discontinuous velocities for the electromagnetic variational method

TL;DR

This paper develops a variational method for electromagnetic two-body problems allowing solutions with discontinuous derivatives, leading to new insights into localized orbits with vanishing far fields and their underlying differential equations.

Contribution

It introduces a generalized variational framework accommodating discontinuous velocities in electromagnetic two-body problems, expanding the classical understanding of solutions and their properties.

Findings

Discontinuous velocities occur at sewing points in localized orbits.

Generalized electromagnetic fields are defined almost everywhere using Li{\'e}nard-Wierchert formulas.

Solutions involve linear neutral differential delay equations for far fields.

Abstract

The electromagnetic two-body problem has \emph{neutral differential delay} equations of motion that, for generic boundary data, can have solutions with \emph{discontinuous} derivatives. If one wants to use these neutral differential delay equations with \emph{arbitrary} boundary data, solutions with discontinuous derivatives must be expected and allowed. Surprisingly, Wheeler-Feynman electrodynamics has a boundary value variational method for which minimizer trajectories with discontinuous derivatives are also expected, as we show here. The variational method defines continuous trajectories with piecewise defined velocities and accelerations, and electromagnetic fields defined \emph{by} the Euler-Lagrange equations \emph{% on} trajectory points. Here we use the piecewise defined minimizers with the Li{\'{e}}nard-Wierchert formulas to define generalized electromagnetic fields almost everywhere (but on sets of points of zero measure where the advanced/retarded velocities and/or accelerations are discontinuous). Along with this generalization we formulate the \emph{generalized absorber hypothesis} that the far fields vanish asymptotically \emph{almost everywhere%} and show that localized orbits with far fields vanishing almost everywhere \emph{must} have discontinuous velocities on sewing chains of breaking points. We give the general solution for localized orbits with vanishing far fields by solving a (linear) neutral differential delay equation for these far fields. We discuss the physics of orbits with discontinuous derivatives stressing the differences to the variational methods of classical mechanics and the existence of a spinorial four-current associated with the generalized variational electrodynamics.