Regular and Chaotic Dynamics

, Volume 19, Issue 2, pp 145–161 | Cite as

Remarks on integrable systems



The problem of integrability conditions for systems of differential equations is discussed. Darboux’s classical results on the integrability of linear non-autonomous systems with an incomplete set of particular solutions are generalized. Special attention is paid to linear Hamiltonian systems. The paper discusses the general problem of integrability of the systems of autonomous differential equations in an n-dimensional space, which admit the algebra of symmetry fields of dimension ⩽ n. Using a method due to Liouville, this problem is reduced to investigating the integrability conditions for Hamiltonian systems with Hamiltonians linear in the momenta in phase space of dimension that is twice as large. In conclusion, the integrability of an autonomous system in three-dimensional space with two independent non-trivial symmetry fields is proved. It should be emphasized that no additional conditions are imposed on these fields.


integrability by quadratures adjoint system Hamiltonian equations Euler — Jacobi theorem Lie theorem symmetries 

MSC2010 numbers



  1. 1.
    Kaplansky, I., An Introduction to Differential Algebra, Paris: Hermann, 1957.MATHGoogle Scholar
  2. 2.
    Kozlov, V.V., The Euler — Jacobi — Lie Integrability Theorem, Regul. Chaotic Dyn., 2013, vol. 18, no. 4, pp. 329–343.CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Olver, P. J., Applications of Lie Groups to Differential Equations, Grad. Texts in Math., vol. 107, New York: Springer, 1986.CrossRefMATHGoogle Scholar
  4. 4.
    Borisov, A.V. and Mamaev, I. S., Modern Methods of the Theory of Integrable Systems, Moscow: R&C Dynamics, ICS, 2003 (Russian).MATHGoogle Scholar
  5. 5.
    Darboux, G., Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal: T. 1, Paris: Gauthier-Villars, 1914.Google Scholar
  6. 6.
    Kozlov, V.V., General Theory of Vortices, Encyclopaedia Math. Sci., vol. 67, Berlin: Springer, 2003.MATHGoogle Scholar
  7. 7.
    Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Lie algebras in vortex dynamics and celestial mechanics: 4, Regul. Chaotic Dyn., 1999, vol. 4, no. 1, pp. 23–50.CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Multiparticle Systems. The Algebra of Integrals and Integrable Cases, Regul. Chaotic Dyn., 2009, vol. 14, no. 1, pp. 18–41.CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Superintegrable system on a sphere with the integral of higher degree, Regul. Chaotic Dyn., 2009, vol. 14, no. 6, pp. 615–620.CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Borisov, A.V. and Mamaev, I. S., Superintegrable systems on a sphere, Regul. Chaotic Dyn., 2005, vol. 10, no. 3, pp. 257–266.CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Borisov, A.V. and Mamaev, I. S., On the problem of motion of vortex sources on a plane, Regul. Chaotic Dyn., 2006, vol. 11, no. 4, pp. 455–466.CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Borisov, A.V. and Pavlov, A. E., Dynamics and statics of vortices on a plane and a sphere: 1, Regul. Chaotic Dyn., 1998, vol. 3, no. 1, pp. 28–38.CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Whittaker, E.T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th ed., New York: Cambridge Univ. Press, 1959.Google Scholar
  14. 14.
    Nekhoroshev, N.N., Action-Angle Variables and Their Generalization, Trans. Moscow Math. Soc., 1972, vol. 26, pp. 180–198; see also: Tr. Mosk. Mat. Obs., 1972, vol. 26, pp. 181–198 (Russian).Google Scholar
  15. 15.
    Mishchenko, A. S. and Fomenko, A. T., Generalized Liouville Method of Integration of Hamiltonian Systems, Funct. Anal. Appl., 1978, vol. 12, no. 2, pp. 113–121; see also: Funktsional. Anal. i Prilozhen., 1978, vol. 12, no. 2, pp. 46–56 (Russian).CrossRefMATHGoogle Scholar
  16. 16.
    Brailov, A. V., Complete Integrability of Some Geodesic Flows and Integrable Systems with Noncommuting Integrals, Dokl. Akad. Nauk SSSR, 1983, vol. 271, no. 2, pp. 273–276 (Russian).MathSciNetGoogle Scholar
  17. 17.
    Stekloff, W., Application du théor`eme généralisé de Jacobi au probl`eme de Jacobi — Lie, C. R. Acad. Sci. Paris, 1909, vol. 148, pp. 465–468.MATHGoogle Scholar
  18. 18.
    Kozlov, V.V., An Extended Hamilton — JacobiMethod, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 580–596.CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Kozlov, V.V., Symmetries, Topology, and Resonances in Hamiltonian Mechanics, Ergeb. Math. Grenzgeb. (3), vol. 31, Berlin: Springer, 1996.CrossRefGoogle Scholar
  20. 20.
    Kozlov, V.V., Remarks on a Lie Theorem on the Exact Integrability of Differential Equations, Differ. Equ., 2005, vol. 41, no. 4, pp. 588–590; see also: Differ. Uravn., 2005, vol. 41, no. 4, pp. 553–555 (Russian).CrossRefMATHMathSciNetGoogle Scholar

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© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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