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Title: Minimizers with discontinuous velocities for the electromagnetic variational method

Abstract

The electromagnetic two-body problem has neutral differential delay equations of motion that, for generic boundary data, can have solutions with discontinuous derivatives. If one wants to use these neutral differential delay equations with 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 by the Euler-Lagrange equations on trajectory points. Here we use the piecewise defined minimizers with the Lienard-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 generalized absorber hypothesis that the far fields vanish asymptotically almost everywhere and show that localized orbits with far fields vanishing almost everywhere 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 differencesmore » to the variational methods of classical mechanics and the existence of a spinorial four-current associated with the generalized variational electrodynamics.« less

Authors:
 [1]
  1. Departamento de Fisica Rodovia Washington Luis, Universidade Federal de Sao Carlos, km 235 Caixa Postal 676, Sao Carlos, Sao Paulo 13565-905, SP (Brazil)
Publication Date:
OSTI Identifier:
21464484
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics (Print); Journal Volume: 82; Journal Issue: 2; Other Information: DOI: 10.1103/PhysRevE.82.026212; (c) 2010 The American Physical Society
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ACCELERATION; CLASSICAL MECHANICS; ELECTRIC CURRENTS; ELECTRODYNAMICS; ELECTROMAGNETIC FIELDS; EQUATIONS OF MOTION; LAGRANGE EQUATIONS; MATHEMATICAL SOLUTIONS; ORBITS; TRAJECTORIES; TWO-BODY PROBLEM; VARIATIONAL METHODS; CALCULATION METHODS; CURRENTS; DIFFERENTIAL EQUATIONS; EQUATIONS; MANY-BODY PROBLEM; MECHANICS; PARTIAL DIFFERENTIAL EQUATIONS

Citation Formats

De Luca, Jayme. Minimizers with discontinuous velocities for the electromagnetic variational method. United States: N. p., 2010. Web. doi:10.1103/PHYSREVE.82.026212.
De Luca, Jayme. Minimizers with discontinuous velocities for the electromagnetic variational method. United States. doi:10.1103/PHYSREVE.82.026212.
De Luca, Jayme. Sun . "Minimizers with discontinuous velocities for the electromagnetic variational method". United States. doi:10.1103/PHYSREVE.82.026212.
@article{osti_21464484,
title = {Minimizers with discontinuous velocities for the electromagnetic variational method},
author = {De Luca, Jayme},
abstractNote = {The electromagnetic two-body problem has neutral differential delay equations of motion that, for generic boundary data, can have solutions with discontinuous derivatives. If one wants to use these neutral differential delay equations with 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 by the Euler-Lagrange equations on trajectory points. Here we use the piecewise defined minimizers with the Lienard-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 generalized absorber hypothesis that the far fields vanish asymptotically almost everywhere and show that localized orbits with far fields vanishing almost everywhere 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.},
doi = {10.1103/PHYSREVE.82.026212},
journal = {Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics (Print)},
number = 2,
volume = 82,
place = {United States},
year = {Sun Aug 15 00:00:00 EDT 2010},
month = {Sun Aug 15 00:00:00 EDT 2010}
}
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