Mathematical models and contemporary theories of physical fields

V. A. Kovalev ; Yu. N. Radayev

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 9 (2009) no. 4, pp. 41-94.

Résumé

Elements of the classical field theory based on a variational formulation of the Hamilton type are discussed and corresponding 4-dimensional Lagrange formalism is presented both as the variational and the group theoretical script. Variational symmetries (geometric and generalized) of field equations and theNoether theoremproviding a regular way of obtaining a conservation law for every given variational symmetry are revisited in the study in order to give a complete version of the contemporary field theory. All developments are presented in the non-linear frame (i.e. of finite strains as to continuum mechanics). Natural derivations of all tensor attributes of a physical field are given by the variational symmetry technique. The null Lagrangian theory for $n$-dimensional manifold (including 4-dimensional Minkowski space-time) is developed in an attempt to extend the canonical formalismof non-linear field theory. By the aid of divergence formula for the null Lagrangians regular in $n$-dimensional star-shaped domains, a general representation of the null Lagrangian depending as maximum on the first order field gradients is obtained. A method of systematic and derivation of the null Lagrangians for $n$-dimensional manifold is proposed. It is shown that in the case of non-linear 3-component field in 3-dimensional space the null Lagrangian is represented, in general, via 15 arbitrary independent field functions.

Bibliographie

[1] Landau L. D., Lifshits E. M., Teoreticheskaya fizika: V 2 t., v. I, Mekhanika, Nauka, M., 1973, 208 pp. | MR

[2] Landau L. D., Lifshits E. M., Teoreticheskaya fizika: V 2 t., v. II, Teoriya polya, Nauka, M., 1973, 504 pp. | MR

[3] Noether E., “Invariante Variationsprobleme”, Kgl. Ges. Wiss. Nachr. Gottingen. Math.-Physik. Kl. 2, 1918, 235–257 | Zbl

[4] Radaev Yu. N., Gudkov V. A., “O vychislenii nulevykh Lagranzhianov nelineino uprugogo polya”, Vestn. Samar. gos. un-ta. Estestvenno-nauch. ser. Spets. vyp., 2002, 39–56 | MR | Zbl

[5] Maugin G. A., Material Inhomogeneities in Elasticity, Chapman Hall, L., 1993, 276 pp. | MR | Zbl

[6] Berdichevskii V. L., Variatsionnye printsipy mekhaniki sploshnoi sredy, Nauka, M., 1983, 448 pp. | MR

[7] Kurant R., Gilbert D., Metody matematicheskoi fiziki: V 2 t., v. 1, Gostekhteoretizdat, M., L., 1933, 528 pp.

[8] Courant R., Hilbert D., Methods of Mathematical Physics, v. 1, Interscience Publishers, N.Y., 1953, 562 pp., Per. na rus. yaz. sm. [6] | MR

[9] Gelfand I. M., Fomin S. V., Variatsionnoe ischislenie, Fizmatgiz, M., 1961, 228 pp. | MR

[10] Gelfand I. M., Fomin S. V., Calculus of Variations., Revised English ed., eds. R. A. Silverman, Prentice-Hall, Englewood Cliffs, New Jersey, 1963, 232 pp., Originalnoe izdanie sm. [9] | MR | Zbl

[11] Ovsyannikov L. V., Gruppovoi analiz differentsialnykh uravnenii, Nauka, M., 1978, 400 pp. | MR

[12] Ibragimov N. Kh., Gruppy preobrazovanii v matematicheskoi fizike, Nauka, M., 1983, 280 pp. | MR

[13] Olver P. J., Application of Lie Groups to Differential Equations, Springer, N.Y., 1986, Per. na rus. yaz. sm. [14] | MR

[14] Olver P., Prilozheniya grupp Li k differentsialnym uravneniyam, Mir, M., 1989, 639 pp. | MR | Zbl

[15] Olver P. J., Equivalence, Invariants and Symmetry, Cambridge University Press, Cambridge, N.Y., Melbourne, 1995 | MR | Zbl

[16] Einshtein A., Sobranie nauchnykh trudov: V 4 t., v. 1, Raboty po teorii otnositelnosti, Nauka, M., 1965, 700 pp.

[17] Dirak P. A. M., Obschaya teoriya otnositelnosti, Atomizdat, M., 1978, 64 pp.

[18] Truesdell C., Toupin R. A., “The Classical Field Theories”, Principles of Classical Mechanics and Field Theory, Part 1, Encyclopedia of Physics, III, ed. S. Flugge, Springer, Berlin, 1960, 226–793 | MR

[19] Lure A. I., Nelineinaya teoriya uprugosti, Nauka, M., 1980, 512 pp. | MR

[20] Silhavy M., The Mechanics and Thermodynamics of Continuous Media, Springer-Verlag, Berlin, Heidelberg, N.Y., 1997, 506 pp. | MR | Zbl

[21] Bessel-Hagen E., “Uber die Erhaultungssatze der Elektrodynamik”, Math. Ann., 84 (1921), 258–276 | DOI | MR | Zbl

[22] Meller K., Teoriya otnositelnosti, Atomizdat, M., 1975, 400 pp.

[23] Maugin G. A., “Material forces: Concepts and applications”, Applied Mechanics Reviews, 48 (1995), 213–245 | DOI | MR

[24] Piola G., “Nuovo analisi per tutti le questioni della meccanica moleculare”, Mem. Mat. Fis. Soc. Ital. Modena, 21 (1835), 155–321

[25] Piola G., “Intorno alle equazioni fondamentali del movimento di corpi qualsivoglioni considerati secondo la naturale loro forma e costituva”, Mem. Mat. Fis. Soc. Ital. Modena, 24(1) (1848), 1–186

[26] Eshelby J. D., “The Force on an Elastic Singularity”, Phil. Trans. Roy. Soc. L. A, 244 (1951), 87–112 | DOI | MR | Zbl

[27] Shvarts L., Analiz: V 2 t., v. II, Mir, M., 1972, 528 pp.

[28] Kartan A., Differentsialnoe ischislenie. Differentsialnye formy, Mir, M., 1971, 392 pp. | MR | Zbl

[29] Mak-Konnel A. Dzh., Vvedenie v tenzornyi analiz s prilozheniyami k geometrii, mekhanike i fizike, Fizmatgiz, M., 1963, 412 pp.