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Variational equations of electrodynamics

Variational equations of electrodynamics ISSN 0012-2661, Differential Equations, 2007, Vol. 43, No. 4, pp. 488–494.  c Pleiades Publishing, Ltd., 2007. Original Russian Text c E.R. Smol’yakov, 2007, published in Differentsial’nye Uravneniya, 2007, Vol. 43, No. 4, pp. 475–480. ORDINARY DIFFERENTIAL EQUATIONS E. R. Smol’yakov Moscow State University, Moscow, Russia Received November 16, 2005 DOI: 10.1134/S0012266107040064 1. INTRODUCTION A variational derivation of the equations of electrodynamics on the basis of the complete “action integral” in the form (28.6) in [1, p. 105] [that is, the integral (1) below] with regard to both the energy of charged particles and the energy of the ambient field itself has not been performed yet. Solutions (in the form of the Maxwell–Lorentz equations) were found only for the pair of variational problems obtained if one splits the original problem with the functional (28.6) in [1, p. 105] to be optimized into two simplified variational problems [1, pp. 72–75, 108–109]. Such an expansion does not take into account some consequences of the interaction of charges with the ambient electromagnetic field (induced by both these charges and external sources). The results obtained so far in the optimization theory of vector and complex functionals in arbitrary vector and complex spaces [2] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

Variational equations of electrodynamics

Smol’yakov, E.
Differential Equations , Volume 43 (4) – May 24, 2007
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References (2)

Publisher
Springer Journals
Copyright
Copyright © 2007 by Pleiades Publishing, Ltd.
Subject
Mathematics; Ordinary Differential Equations; Partial Differential Equations; Difference and Functional Equations
ISSN
0012-2661
eISSN
1608-3083
DOI
10.1134/S0012266107040064
Publisher site
See Article on Publisher Site

Abstract

ISSN 0012-2661, Differential Equations, 2007, Vol. 43, No. 4, pp. 488–494.  c Pleiades Publishing, Ltd., 2007. Original Russian Text c E.R. Smol’yakov, 2007, published in Differentsial’nye Uravneniya, 2007, Vol. 43, No. 4, pp. 475–480. ORDINARY DIFFERENTIAL EQUATIONS E. R. Smol’yakov Moscow State University, Moscow, Russia Received November 16, 2005 DOI: 10.1134/S0012266107040064 1. INTRODUCTION A variational derivation of the equations of electrodynamics on the basis of the complete “action integral” in the form (28.6) in [1, p. 105] [that is, the integral (1) below] with regard to both the energy of charged particles and the energy of the ambient field itself has not been performed yet. Solutions (in the form of the Maxwell–Lorentz equations) were found only for the pair of variational problems obtained if one splits the original problem with the functional (28.6) in [1, p. 105] to be optimized into two simplified variational problems [1, pp. 72–75, 108–109]. Such an expansion does not take into account some consequences of the interaction of charges with the ambient electromagnetic field (induced by both these charges and external sources). The results obtained so far in the optimization theory of vector and complex functionals in arbitrary vector and complex spaces [2]

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Differential Equations – Springer Journals

Published: May 24, 2007

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