First-order covariant differential operators

Yu. P. Virchenko ; A. V. Subbotin

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry and Mechanics, Tome 187 (2020), pp. 19-30

Résumé

An internal description of the class of all nonlinear differential operators of the first order on the space of collections consisting of continuously differentiable vector and scalar fields on $\mathbb{R}^3$ is given. Operators of this class are invariant with respect to translations of $\mathbb{R}^3$ and are transformed by the covariant way under rotations of $\mathbb{R}^3$.

Détail

Keywords: first-order differential operator, divergence differential operator, vector field, pseudo-vector field, covariance.

Bibliographie

[6] Andreev A. F., Marchenko V. I., “Makroskopicheskaya teoriya spinovykh voln”, ZhETF., 70:4 (1976), 1522–1532

[7] Andreev A. F., Marchenko V. I., “Simmetriya i makroskopicheskaya dinamika magnetikov”, Usp. fiz. nauk., 130 (1) (1980), 37–63

[8] Virchenko Yu. P., Peletminskii S. V., “Skobki Puassona i differentsialnye zakony sokhraneniya v teorii magnitouprugikh sred”, Problemy fizicheskoi kinetiki i fiziki tverdogo tela, Naukova dumka, Kiev, 1990, 63–77

[9] Volkov D. V., “Fenomenologicheskie lagranzhiany”, Fiz. el. chast. atom. yadra., 4:1 (1973), 3–41 | MR

[10] Volkov D. V., Zheltukhin A. A., “Fenomenologicheskii lagranzhian spinovykh voln v prostranstvenno-neuporyadochennykh sredakh”, Fiz. nizk. temp., 5:11 (1979), 1359–1363

[11] Volkov D. V., Zheltukhin A. A., Bliokh Yu. P., “Fenomenologicheskii lagranzhian spinovykh voln”, Fiz. tv. tela., 13:6 (1971), 1668–1678

[12] Volovik G. E., Kats E. I., “Nelineinaya gidrodinamika zhidkikh kristallov”, ZhETF., 81 (1981), 240–248

[13] Gurevich G. B., Osnovy teorii algebraicheskikh invariantov, GITTL, M.-L., 1948 | MR

[14] Lyubarskii G. Ya., Teoriya grupp i ee prilozheniya v fizike, GIFML, M., 1958

[15] Ponamareva A. E., Virchenko Yu. P., “Postroenie obschego evolyutsionnogo uravneniya dlya psevdovektornogo solenoidalnogo polya s lokalnym zakonom sokhraneniya”, Nauch. ved. Belgorod. un-ta. Ser. Mat. Fiz., 50:2 (2018), 224–232

[16] Subbotin A. V., “Opisanie klassa evolyutsionnykh uravnenii divergentnogo tipa dlya vektornogo polya”, Nauch. ved. Belgorod. un-ta. Ser. Mat. Fiz., 50:4 (2018), 492–497

[17] Dieudonne J. A. Carrell J. A., Invariant theory. Old and new, Academic Press, New York, 1971 | MR | Zbl

[18] Dzyaloshinskii I. E., “Macroscopic description of spin glasses”, Lect. Notes Phys., 115 (1980), 204–224 | DOI | MR

[19] Dzyaloshinskii I. E., Volovick G. E., “Poisson brackets in condensed matter physics”, Ann. Phys., 125:1 (1980), 67–97 | DOI | MR

[20] Golo V. L., Monastyrsky M. I., Novikov S. P., “Solutions to the Ginzburg–Landau equations for planar textures in superfluid ${}^3$He”, Commun. Math. Phys., 69:3 (1979), 237–246 | DOI | MR

[21] Halperin B. I., Hohenberg P. C., “Hydrodynamic theory of spin waves”, Phys. Rev., 88:2 (1969), 898–919 | DOI

[22] Leggett A. J., “A theoretical description of the new phases of liquid ${}^3$He”, Rev. Mod. Phys., 47 (1975), 331–414 | DOI

[23] McConnel A. J., Application of tensor analysis, Dover, New York, 1957 | MR | Zbl

[24] Spencer A. G. M., “Theory of Invariants”, Continuum Physics, eds. Eringen A. C., Academic Press, New York, 1971, 239–353 | MR

[25] Volovik G. E., “Relationship between molecule shape and hydrodynamics in a nematic substance”, JETP Lett., 31:5 (1980), 273–275