Advertisement

Siberian Mathematical Journal

, Volume 47, Issue 2, pp 258–268 | Cite as

Invariant tensors and partial differential equations

  • O. V. Kaptsov
Article
  • 30 Downloads

Abstract

We consider tensors with coefficients in a commutative differential algebra A. Using the Lie derivative, we introduce the notion of a tensor invariant under a derivation on an ideal of A. Each system of partial differential equations generates an ideal in some differential algebra. This makes it possible to study invariant tensors on such an ideal. As examples we consider the equations of gas dynamics and magnetohydrodynamics.

Keywords

integral invariants Lie derivative invariant tensor 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Poincaré H., “New methods of celestial mechanics,” in: Selected Works [Russian translation], Moscow, Nauka, 1972, pp. 9–445.Google Scholar
  2. 2.
    Arnol’d V. I., Mathematical Methods in Classical Mechanics [in Russian], Nauka, Moscow (1979).Google Scholar
  3. 3.
    Kozlov V. V., General Theory of Vortices [in Russian], RKhD, Izhevsk (2000).Google Scholar
  4. 4.
    Cartan É., Integral Invariants [Russian translation], Gostekhizdat, Moscow; Leningrad (1940).Google Scholar
  5. 5.
    Godbillon C., Differential Geometry and Analytic Mechanics [Russian translation], Mir, Moscow (1973).Google Scholar
  6. 6.
    Kobayashi Sh. and Nomizu K., Foundations of Differential Geometry [Russian translation], Nauka, Moscow (1981).Google Scholar
  7. 7.
    Bourbaki N., Algebra (Polynomials and Fields. Ordered Groups) [Russian translation], Nauka, Moscow (1965).Google Scholar
  8. 8.
    Kolchin E. A., Differential Algebra and Algebraic Groups, Acad. Press, New York; London (1973).Google Scholar
  9. 9.
    Goursat E., Lecons sur le probleme de Pfaff, Herman, Paris (1922).Google Scholar
  10. 10.
    Kochin N. E., Kibel’ I. A., and Rose N. V., Theoretical Hydrodynamics [in Russian], Fizmatlit, Moscow (1963).Google Scholar
  11. 11.
    Courant R., Partial Differential Equations [Russian translation], Mir, Moscow (1964).Google Scholar
  12. 12.
    Serrin J. B., Mathematical Principles of Classical Fluid Mechanics [Russian translation], Izdat. Inostr. Lit., Moscow (1963).Google Scholar
  13. 13.
    Landau L. D. and Lifshits E. M., Electrodynamics of Continuous Media [in Russian], Nauka, Moscow (2001).Google Scholar
  14. 14.
    Holm D. D., Marsden J. E., Ratiu T., and Weinstein A., “Nonlinear stability of fluid and plasma equilibria,” Phys. Reports, 123, No. 1&2, 1–116 (1985).MathSciNetGoogle Scholar
  15. 15.
    Kirillov A. A., Lectures on the Orbit Method [in Russian], Nauchnaya Kniga, Novosibirsk (2002).Google Scholar
  16. 16.
    Sokolowski J. and Khludnev A. M., “The derivative of the energy functional along the crack length in elasticity problems,” Prikl. Mat. i Mekh., 64, No. 3, 467–475 (2000).Google Scholar
  17. 17.
    Kovtunenko V. A., “The invariant energy integral for a nonlinear crack problem with a possible contact of crack faces,” Prikl. Mat. i Mekh., 67, No. 1, 109–123 (2003).zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • O. V. Kaptsov
    • 1
  1. 1.Institute of Computational ModelingKrasnoyarskRussia

Personalised recommendations