Axiomatic construction of classical electrodynamics equations

The classical electrodynamics equations for a stationary isotropic medium without polarization and magnetization effects with regard to possibility of collective motion for electric charges (vacuum), in contrast to traditional using the least action principle of classical theory of gage fields, are obtained from the postulate that alternating electromagnetic field can be described by a special relativity theory with two vector fields in Minkowski space (4-potential and 4-current). The condition of gage invariance for "power" vector fields in the classical electrodynamics defines antisymmetry of the 4-tensor electromagnetic field. The existence of two specific mathematical structures (pseudo- and true vector in the three-dimensional space), whose components are the ones of the 4-tensor electromagnetic field, is determined. The postulate about two different "power" vector fields of the classical electrodynamics is a natural consequence of two different mathematical structures in the electromagnetic field tensor. The first pair of Maxwell equations (homogeneous equations) is obtained as a consequence offormal definition of corresponding vector fields. The second pair ofMaxwell equations (inhomogeneous equations) is obtained as a consequence ofthe postulate that 4-current vector field is the vector "source" of the 4-tensor electromagnetic field. Vanishing of 4-divergence 4-current is shown to be the necessary condition of the theory concerned. The physical meaning of formally introduced "power" vector fields is determined. The fundamental equations of classical electrodynamics (Ampere’s circuital law and electromagnetic induction law) in the proposed approach are the consequence of the above-mentioned postulates.

References

[1] Sir Edmund Whittaker f.r.s. A History of the Theories of Aether and Electricity. The Classical Theories. London, Edinburg, Paris, Melbourne, Toronto and New York, Thomas Nelson and Sons Ltd., 1953.

[2] Nugaev R.M. The genesis of maxwellian electrodynamics: the inter-theoretic context. Filosofiya nauki [Philosophy of Sciences], 2014, no. 2 (61), pp. 66-80 (in Russ.).

[3] Landau L.D., Lifshits E.M. Course of theoretical physics. Ten-volume set. Vol. 2. The Classical Theory of Fields. Oxford, New York, Pergamon, 1984. 460 p. (Russ. ed.: Landau L.D., Lifshits E.M. Teoreticheskaya fizika. V 10 t. T. 2. Teoriya polya. Moscow, Fizmatlit Publ., 2012. 536 p.).

[4] Landau L.D., Lifshits E.M. Teoreticheskaya fizika. V 10 t. T. 8. Elektrodinamika sploshnykh sred [Theoretical Physics. Vol. 8. Electrodynamics of Continuous Media]. Moscow, Fizmatlit Publ., 2005. 656 p.

[5] Fock V. The Theory of Space Time & Gravitation. Pergamon Press Ltd., 1964 (Russ. ed.: Fok V.A. Teoriya prostranstva, vremeni i tyagoteniya. Moscow, GIFML Press, 1961. 563 p.).

[6] Savelyev I.V. Osnovy teoreticheskoy fiziki. V 2 t. T 1. Mekhanika i elektrodinamika [Fundamentals of Theoretical Physics. In 2 vol. Vol. 1: Mechanics and Electrodynamics]. Moscow, Nauka Publ., 1991. 496 p.

[7] Ugarov V.A. Spetsial’naya teoriya otnositel’nosti [Special Theory of Relativity]. Moscow, Editorial URSS Publ., 2005. 384 p.

[8] Korenev G.V. Tenzornoe ischislenie [Tensor calculus]. Moscow, MFTI Publ., 1995. 240 p.

[9] Rubakov V.A. Klassicheskie kalibrovochnye polya [Classical Theory of Gauge Fields]. Moscow, Editorial URSS Publ., 1999. 335 p.

[10] Makarov A.M., Luneva L.A., Makarov K.A. Teoriya i praktika klassicheskoy elektrodinamiki [Theory and Practice of Classical Electrodynamics]. Moscow, Editorial URSS Publ., 2014. 784 p.

[11] Fushchych W.I., Nikitin A.G. On the new symmetries of Maxwell equations. W.I. Fushchych, Scientific Works, 2000, vol. 2, pp. 165-169 (Russ. ed.: Fushich V.I., Nikitin A.G. Simmetriya uravneniya Maksvella. Kiev: Naukova dumka Publ., 1983. 200 p.).

[12] Makarov A.M., Lunyova L.A., Makarov K.A. Concerning Structure of Simultaneous Equations of the Classical Electrodynamics. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2014, no. 3, pp. 39-52 (in Russ.).

[13] Mansuripur M. On the Foundational Equations of the Classical Electrodynamics. Resonance, 2013, no. 2, pp. 130-150.

[14] Ponomarev Yu.I. The Elusion of Maxwell Equations from the State Function. Charge State Function and Its Connection with the Charge Conservation Law. Uspekhi sovremennogo estestvoznaniya [Advances in Current Natural Sciences], 2009, no. 1, pp. 6-9.

[15] Lutfullin M. Symmetry Reduction of Nonlinear Equations of Classical Electrodynamics. Symmetry in Nonlinear Mathematical Physics, 1997, vol. 1.

[16] Lu Q.Z., Norris S., Su Q., Grobe R. Self-interactions as Predicted by the Dirac -Maxwell Equations. Phys. Rev. A, 2014, vol. 90, p. 034101.

[17] Etkin V.A. Energodynamic Derivation of Maxwell’s Equations. Dokl. nezavisimykh avtorov. Ser. Fizika i astronomiya [The Papers of Independent Authors, physics, astronomy], 2013, iss. 23, pp. 165-174 (in Russ.).

[18] Etkin V.A. Thermodynamic Derivation of Maxwell’s Equations. Available at: http://www.etkin.iri-as.org/napravlen/09elektr/Termod%20vyvod%20uravn%20Maxvela.pdf (accessed 15.05.2015).

[19] Planck Max. Introduction to Theoretical Physics: Theory of electricity and magnetism, vol. 3. Macmillan, 1932. 247 p. (Russ. ed.: Plank M. Vvedenie v teoreticheskuyu fiziku. Teoriya elektrichestva i magnetizma. Moscow, URSS Publ., 2004. 184 p.).

[20] Sindelka M. Derivation of Coupled Maxwell - Schredinger Equations Describe Matter-laser Interaction from First Principles of Quantum Electrodynamics. Phys. Rev. A, 2010, vol. 81, p. 033833.

[21] Vorontsov A.S., Kozlov V.I., Markov M.B. On the Maxwell’s equations in the self-time. Preprint, Inst. Appl. Math., Russian Academy of Science. Available at: http://keldysh.ru/papers/2005/prep28/prep2005_28.html (accessed 05.05.2015).

[22] Kulyabov D.S., Korolkova A.V., Sevastyanov L.A. The Simplest Geometrization of Maxwell’s Equations. Vestn. RUDN. Ser. Matematika, informatika, fizika, 2014, no. 2, pp. 115-172 (in Russ.).

[23] Darrigol O. James MacCullagh‘s Ether: an Optical route to Maxwell Equations? Eur. Phys. J. H., 2010, vol. 35, pp. 133-172. DOI: 10.1140/epjh/e2010-00009-3

[24] Kusnetsov I.V., Zotov K.H. Improving Accuracy of Positioning Mobile Station based on the Calculation of Static Parameters Electromagnetic Field with Maxwell’s Equations. Electrical and Data Processing Facilities System, 2013, vol. 9, no. 1, pp. 89-92.

[25] Galev R.V., Kovalev O.B. About the Use Maxwell Equations in Numerical Simulation of Interaction of Laser Radiation with Materials. VestnikNGU. Ser. Fizika [Vestnik of NSU: Physics Series], 2014, vol. 9, pp. 53-64 (in Russ.).

[26] Alekseev G.V., Brizitskiy R.V. Theoretical analysis of boundary control extremal problems for Maxwell’s equations. Sib. Zh. Ind. Mat., 2011, vol. 14, no. 1 (45), pp. 3-16 (in Russ.).

[27] Barbas A., Velarde P. Development of a Godunov Method for Maxwell’s Equations with Adaptive Mesh Refinement. Journal of Computational Physics, 2015, vol. 300, pp. 188-201. DOI: 10.1016/j.jcp.2015.07.048

[28] Markel V., Schotland J.C. Homogenization of Maxwell’s Equations in Periodic Composites: Boundary Effects and Dispersion Relation. Phys. Rev. E, 2012, vol. 85, pp. 066603-1-066603-23. Available at: http://journals.aps.org/pre/pdf/10.1103/PhysRevE.85.066603 DOI: 10.1103/PhysRevE.85.066603

[29] Petrov E.Yu., Kudrin A.V. Exact Axisymmetric Solution of the Maxwell Equations in a Nonlinear Nondispersive Medium. Phys. Rev. Lett., 2010, vol. 104, p. 190404. Available at: http://arxiv.org/pdf/1306.0874v1.pdf DOI: 10.1103/PhysRevLett.104.190404