Regular and Chaotic Dynamics

, Volume 18, Issue 4, pp 329–343 | Cite as

The Euler-Jacobi-Lie integrability theorem

  • Valery V. Kozlov


This paper addresses a class of problems associated with the conditions for exact integrability of systems of ordinary differential equations expressed in terms of the properties of tensor invariants. The general theorem of integrability of the system of n differential equations is proved, which admits n − 2 independent symmetry fields and an invariant volume n-form (integral invariant). General results are applied to the study of steady motions of a continuum with infinite conductivity.


symmetry field integral invariant nilpotent group magnetic hydrodynamics 

MSC2010 numbers



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  1. 1.
    Arnold, V. I., Kozlov, V.V., and Neîshtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 1993, pp. 1–291.CrossRefGoogle Scholar
  2. 2.
    Olver, P. J., Applications of Lie Groups to Differential Equations, 2nd ed., Grad. Texts in Math., vol. 107, New York: Springer, 1993.zbMATHCrossRefGoogle Scholar
  3. 3.
    Kozlov, V.V., Symmetries, Topology and Resonances in Hamiltonian Mechanics, Ergeb. Math. Grenzgeb. (3), vol. 31, Berlin: Springer, 1996.CrossRefGoogle Scholar
  4. 4.
    Kozlov, V.V., Remarks on a Lie Theorem on the Exact Integrability of Differential Equations, Differ. Uravn., 2005, vol. 41, no. 4, pp. 553–555, 576 [Differ. Equ., 2005, vol. 41, no. 4, pp. 588–590].MathSciNetGoogle Scholar
  5. 5.
    Kozlov, V.V., On the Theory of Integration of the Equations of Nonholonomic Mechanics, Adv. in Mech., 1985, vol. 8, no. 3, pp. 85–107 (Russian).MathSciNetGoogle Scholar
  6. 6.
    Bolotin, S. V. and Kozlov, V. V., Symmetry Fields of Geodesic Flows, Russ. J. Math. Phys., 1995, vol. 3, no. 3, pp. 279–295.MathSciNetzbMATHGoogle Scholar
  7. 7.
    Kozlov, V.V., Symmetries and Regular Behavior of Hamiltonian Systems, Chaos, 1996, vol. 6, no. 1, pp. 1–5.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Pesin, Ya.B., Characteristic Lyapunov Exponents and Smooth Ergodic Theory, Uspekhi Mat. Nauk, 1977, vol. 32, no. 4(196), pp. 55–112 [Russian Math. Surveys, 1977, vol. 32, no. 4, pp. 55–114].MathSciNetGoogle Scholar
  9. 9.
    Sedov, L. I., A Course in Continuum Mechanics: Basic Equations and Analytical Techniques, Groningen: Wolters-Noordhoff, 1971.zbMATHGoogle Scholar
  10. 10.
    Kozlov, V.V., Notes on Steady Vortex Motions of Continuous Medium, Prikl. Mat. Mekh., 1983, vol. 47, no. 2, pp. 341–342 [J. Appl. Math. Mech., 1983, vol. 47, no. 2, pp. 288–289].MathSciNetGoogle Scholar
  11. 11.
    Arnol’d, V. I., On the Topology of Three-Dimensional Steady Flows of an Ideal Fluid, Prikl. Mat. Mekh., 1966, vol. 30, no. 1, pp. 183–185 [J. Appl. Math. Mech., 1966, vol. 30, no. 1, pp. 223–226].Google Scholar
  12. 12.
    Borisov, A.V. and Mamaev, I. S., Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems, Regul. Chaotic Dyn., 2008, vol. 13, no. 5, pp. 443–490.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Hamiltonization of Nonholonomic Systems in the Neighborhood of Invariant Manifolds, Regul. Chaotic Dyn., 2011, vol. 16, no. 5, pp. 443–464.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support, Regul. Chaotic Dyn., 2012, vol. 17, no. 2, pp. 170–190.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Kim, B., Routh Symmetry in the Chaplygin’s Rolling Ball, Regul. Chaotic Dyn., 2011, vol. 16, no. 6, pp. 663–670.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Kozlov, V.V., On Invariant Manifolds of Nonholonomic Systems, Regul. Chaotic Dyn., 2012, vol. 17, no. 2, pp. 131–141.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.V. A. Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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