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Siberian Mathematical Journal

, Volume 60, Issue 3, pp 508–515 | Cite as

Lie-Admissible Algebras Associated with Dynamical Systems

  • V. M. SavchinEmail author
  • S. A. BudochkinaEmail author
Article
  • 19 Downloads

Abstract

We introduce the general structures of Lie-admissible algebras in the spaces of Gâteaux differentiable operators and establish their connection with the symmetries of operator equations and the mechanics of infinite-dimensional systems.

Keywords

Lie-admissible algebra Lie algebra \((\mathscr{S}, \mathscr{T})\)-product \(\mathscr{G}\)-commutator symmetry Gâteaux derivative recursion operator 

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References

  1. 1.
    Santilli R. M., “Mathematical studies on Lie-admissible algebras,” Hadronic J., vol. 1, 574–901 (1977).Google Scholar
  2. 2.
    Santilli R. M., Foundations of Theoretical Mechanics. II. Birkhoffian Generalization of Hamiltonian Mechanics, Springer-Verlag, New York (1983).zbMATHGoogle Scholar
  3. 3.
    Borisov A. V. and Mamaev I. S., Poisson Structures and Lie Algebras in Hamiltonian Mechanics [Russian], Udmurt. Univ.; Regulyarnaya i Khaoticheskaya Dinamika, Izhevsk (1999).zbMATHGoogle Scholar
  4. 4.
    Kozlov V. V., General Theory of Vortices [Russian], Institute of Computer Science, Moscow and Izhevsk (2013).Google Scholar
  5. 5.
    Savchin V. M., Mathematical Methods of Mechanics of Infinite-Dimensional Nonpotential Systems [Russian], RUDN University, Moscow (1991).zbMATHGoogle Scholar
  6. 6.
    Savchin V. M., “On the structure of a Lie-admissible algebra in the space of Gateaux differentiable operators,” Math. Notes, vol. 55, no. 1, 103–104 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Lyusternik L. A. and Sobolev V. I., Elements of Functional Analysis [Russian], Nauka, Moscow (1965).zbMATHGoogle Scholar
  8. 8.
    Savchin V. M. and Budochkina S. A., “On connection between symmetries of functionals and equations,” Dokl. Math., vol. 90, no. 2, 626–627 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Savchin V. M. and Budochkina S. A., “Invariance of functionals and related Euler-Lagrange equations,” Russian Math. (Izv. VUZ), vol. 62, no. 2, 49–54 (2017).CrossRefzbMATHGoogle Scholar
  10. 10.
    Savchin V. M., “Symmetries of PDEs with deviating arguments,” in: Proceedings of the XXXVII All-Russia Scientific Conference on the Problems of Mathematics, Informatics, Physics, Chemistry, and Methods of Teaching Natural Sciences, RUDN University, Moscow, 2001, 7–8.Google Scholar
  11. 11.
    Olver P., Applications of Lie Groups to Differential Equations, Springer-Verlag, New York (1986).CrossRefzbMATHGoogle Scholar
  12. 12.
    Filippov V. M., Savchin V. M., and Shorokhov S. G., “Variational principles for nonpotential operators,” J. Math. Sci., vol. 68, no. 3, 275–398 (1994).MathSciNetCrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Inc. 2019

Authors and Affiliations

  1. 1.Peoples’ Friendship University of Russia (RUDN University) S. M. Nikolskii Mathematical InstituteMoscowRussia

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