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S. Anco, G. Bluman (1997)
Direct Construction of Conservation Laws from Field EquationsPhysical Review Letters, 78
E. Cunningham
The Principle of Relativity in Electrodynamics and an Extension ThereofProceedings of The London Mathematical Society
D. Fairlie (1965)
Conservation laws and invariance principlesIl Nuovo Cimento (1955-1965), 37
P. Olver (1986)
Applications of lie groups to differential equationsActa Applicandae Mathematica, 20
(1995)
in Advanced Electromagnetism: Foundations
S. Anco, G. Bluman (1997)
Nonlocal symmetries and nonlocal conservation laws of Maxwell’s equationsJournal of Mathematical Physics, 38
W. Dietz, R. Rüdiger (1980)
Shearfree congruences of null geodesies and killing tensorsGeneral Relativity and Gravitation, 12
P. Clarkson (1993)
Applications of analytic and geometric methods to nonlinear differential equations
H. Bateman
The Transformation of the Electrodynamical EquationsProceedings of The London Mathematical Society
D. Lipkin (1964)
Existence of a New Conservation Law in Electromagnetic TheoryJournal of Mathematical Physics, 5
I. Krasilshchik, A. Vinogradov (1984)
Nonlocal symmetries and the theory of coverings: An addendum to A. M. vinogradov's ‘local symmetries and conservation laws”Acta Applicandae Mathematica, 2
S. Anco (2003)
Conservation laws of scaling-invariant field equationsJournal of Physics A, 36
E. Bessel-Hagen (1921)
Über die Erhaltungssätze der ElektrodynamikMathematische Annalen, 84
(2004)
CRM Proceedings and Lecture Notes 34
G. Bluman, S. Anco (2002)
Symmetry and Integration Methods for Differential Equations
(1992)
Contemp
G. Bluman, G. Reid, S. Kumei (1988)
New classes of symmetries for partial differential equationsJournal of Mathematical Physics, 29
Edward Kenschaft (2008)
Master's Thesis
(1997)
lectures given at the Second Mexican School on Gravitation and Mathematical Physics held in Tlaxcala
T. Morgan (1964)
Two Classes of New Conservation Laws for the Electromagnetic Field and for Other Massless FieldsJournal of Mathematical Physics, 5
(1965)
Nuovo Cimento 37
(1996)
Commun
W. Dietz, R. Rüdiger (1981)
Space–times admitting Killing–Yano tensors. IProceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 375
S. Anco, G. Bluman (2001)
Direct construction method for conservation laws of partial differential equations Part II: General treatmentEuropean Journal of Applied Mathematics, 13
W. Fushchich, A. Nikitin (1992)
The complete sets of conservation laws for the electromagnetic fieldJournal of Physics A, 25
J. Pohjanpelto (1995)
SYMMETRIES, CONSERVATION LAWS, AND MAXWELL'S EQUATIONS
(1984)
Acta Appl
G. Bluman (1993)
Potential Symmetries and Linearization
G. Bluman, Patrick Doran-Wu (1995)
The use of factors to discover potential systems or linearizationsActa Applicandae Mathematica, 41
I. Anderson, C. Torre (1994)
Classification of local generalized symmetries for the vacuum Einstein equationsCommunications in Mathematical Physics, 176
W. Fushchich, A. Nikitin, D. Kobe (1987)
Symmetries of Maxwell’s Equations
S. Anco, G. Bluman (1996)
Derivation of conservation laws from nonlocal symmetries of differential equationsJournal of Mathematical Physics, 37
(2003)
A: Math
S. Anco, J. Pohjanpelto (2001)
Classification of Local Conservation Laws of Maxwell's EquationsActa Applicandae Mathematica, 69
I. Krasil’shchik, Alexandre Vinogradov (1989)
Nonlocal trends in the geometry of differential equations: Symmetries, conservation laws, and Bäcklund transformationsActa Applicandae Mathematica, 15
T. Kibble (1965)
Conservation Laws for Free FieldsJournal of Mathematical Physics, 6
S. Anco, J. Pohjanpelto (2002)
Conserved currents of massless fields of spin s ≥ ½Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 459
New nonlocal symmetries and conservation laws are derived for Maxwell's equations in 3 + 1 dimensional Minkowski space using a covariant system of joint vector potentials for the electromagnetic tensor field and its dual. A key property of this system, as well as of this class of new symmetries and conservation laws, is their invariance under the duality transformation that exchanges the electromagnetic field with its dual. (In contrast the standard potential system using a single vector potential is not duality-invariant.) The nonlocal symmetries of Maxwell's equations come from an explicit classification of all symmetries of a certain natural geometric form admitted by the joint potential system in Lorentz gauge. In addition to scaling and duality-rotation symmetries, and the well-known Poincaré and dilation symmetries which involve homothetic Killing vectors, the classification yields new geometric symmetries involving Killing–Yano tensors related to rotations/boosts and inversions. The nonlocal conservation laws of Maxwell's equations are constructed from these geometric symmetries by applying a conserved current formula that uses the joint potentials and directly generates conservation laws from any (local or nonlocal) symmetries of Maxwell's equations. This formula is shown to arise through a series of mappings that relate, respectively, symmetries/adjoint-symmetries of the joint potential system and adjoint-symmetries/symmetries of Maxwell's equations. The mappings are derived as by-products of the study of cohomology of closed one-forms and two-forms locally constructed from the electromagnetic field and its derivatives to any finite order for all solutions of Maxwell's equations. In particular it is shown that the only nontrivial cohomology consists of the electromagnetic field (two-form) itself as well as its dual (two-form), and that this two-form cohomology is killed by the introduction of corresponding potentials.
Acta Applicandae Mathematicae – Springer Journals
Published: Feb 9, 2006
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