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Solutions of the Wheeler-Feynman equations with discontinuous velocities

### Abstract

We generalize Wheeler-Feynman electrodynamics with a variational boundary value problem for continuous boundary segments that might include velocity discontinuity points. Critical-point orbits must satisfy the Euler-Lagrange equations of the action functional at most points, which are neutral differential delay equations (the Wheeler-Feynman equations of motion). At velocity discontinuity points, critical-point orbits must satisfy the Weierstrass-Erdmann continuity conditions for the partial momenta and the partial energies. We study a special setup having the shortest time-separation between the (infinite-dimensional) boundary segments, for which case the critical-point orbit can be found using a two-point boundary problem for an ordinary differential equation. For this simplest setup, we prove that orbits can have discontinuous velocities. We construct a numerical method to solve the Wheeler-Feynman equations together with the Weierstrass-Erdmann conditions and calculate some numerical orbits with discontinuous velocities. We also prove that the variational boundary value problem has a unique solution depending continuously on boundary data, if the continuous boundary segments have velocity discontinuities along a reduced local space.

© 2015 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License.

Received 15 April 2014
Accepted 17 December 2014
Published online 06 January 2015

Lead Paragraph:
The gravitational two-body problem has a unique solution with given initial positions and velocities, an initial value problem for an ordinary differential equation (ODE) in the *finite-dimensional* space
**. Surprisingly, the electromagnetic two-body problem needs an infinite-dimensional initial condition to determine a unique solution, just like a partial differential equation (PDE), e.g., the scalar wave equation with initial function at ***t*
** **
**=**
** **
**0. In that respect PDE's are similar to differential-delay equations with constant delay (DDE). A PDE, either linear or nonlinear, is a differential problem using ***local* information of the infinite-dimensional domain space, i.e., the partial derivatives. State-dependent delay makes the electromagnetic problem harder than a PDE because the functional problem uses non-local information from the infinite-dimensional domain space. Our problem at hand is then harder than a nonlinear PDE. Furthermore, there is a similarity with a shooting problem for a non-autonomous ODE, only that the shooting is in the infinite-dimensional domain space, i.e., starts from the past history segment and lands on the future history segment. Last, the presence of neutral delay introduces difficulties unseen with ODE's, i.e., at every breaking point trajectories pick a velocity discontinuity that does not smooth away, thus creating natural chains of velocity discontinuities.

Acknowledgments:
Daniel Câmara de Souza acknowledges the support of a FAPESP scholarship, proc. 2010/16964-0 and Jayme De Luca acknowledges the partial support of a FAPESP regular Grant, proc. 2011/18343-6.

Article outline:

I. INTRODUCTION
II. VARIATIONAL BOUNDARY VALUE PROBLEM
III. SHOOTING PROBLEM AND WEIERSTRASS-ERDMANN CORNER CONDITIONS
IV. NUMERICAL EXPERIMENTS
V. DISCUSSIONS AND REMARKS

/content/aip/journal/chaos/25/1/10.1063/1.4905201

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http://dx.doi.org/10.1016/j.cam.2012.02.039
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2015-01-06

2016-10-24

### Abstract

We generalize Wheeler-Feynman electrodynamics with a variational boundary value problem for continuous boundary segments that might include velocity discontinuity points. Critical-point orbits must satisfy the Euler-Lagrange equations of the action functional at most points, which are neutral differential delay equations (the Wheeler-Feynman equations of motion). At velocity discontinuity points, critical-point orbits must satisfy the Weierstrass-Erdmann continuity conditions for the partial momenta and the partial energies. We study a special setup having the shortest time-separation between the (infinite-dimensional) boundary segments, for which case the critical-point orbit can be found using a two-point boundary problem for an ordinary differential equation. For this simplest setup, we prove that orbits can have discontinuous velocities. We construct a numerical method to solve the Wheeler-Feynman equations together with the Weierstrass-Erdmann conditions and calculate some numerical orbits with discontinuous velocities. We also prove that the variational boundary value problem has a unique solution depending continuously on boundary data, if the continuous boundary segments have velocity discontinuities along a reduced local space.

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