Doklady Mathematics

, Volume 95, Issue 3, pp 211–213 | Cite as

Liouville nonintegrability of sub-Riemannian problems on free Carnot groups of step 4



One of the main approaches to the study of the Carnot–Carathéodory metrics is the Mitchell–Gromov nilpotent approximation theorem, which reduces the consideration of a neighborhood of a regular point to the study of the left-invariant sub-Riemannian problem on the corresponding Carnot group. A detailed analysis of sub-Riemannian extremals is usually based on the explicit integration of the Hamiltonian system of Pontryagin’s maximum principle. In this paper, the Liouville nonintegrability of this system for left-invariant sub-Riemannian problems on free Carnot groups of step 4 and higher is proved.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. T. Fomenko and A. V. Bolsinov, Integrable Hamiltonian Systems: Geometry, Topology, Classification (Taylor & Francis, London, 2003).MATHGoogle Scholar
  2. 2.
    R. Brockett, in New Directions in Applied Mathematics, Ed. by P. Hilton and G. Young (Springer, New York, 1982), pp. 11–27.CrossRefGoogle Scholar
  3. 3.
    R. Brockett and L. Dai, in Nonholonomic Motion Planning, Ed. by Z. Li and J. Canny (Kluwer, Dordrecht, 1993), pp. 1–21.CrossRefGoogle Scholar
  4. 4.
    B. Gaveau, Acta Math. 139, 95–153 (1977).MathSciNetCrossRefGoogle Scholar
  5. 5.
    A. Anzaldo Meneses and F. Monroy Pérez, J. Math. Phys. 44 (12), 6101–6111 (2003).MathSciNetCrossRefGoogle Scholar
  6. 6.
    O. Myasnichenko, J. Dyn. Control Syst. 8, 573–597 (2002).MathSciNetCrossRefGoogle Scholar
  7. 7.
    J. P. Gauthier and Yu. L. Sachkov, Program. Sist. Teor. Prilozh. 6 (2), 45–61 (2015).Google Scholar
  8. 8.
    A. A. Kirillov, Lectures on the Orbit Method (Nauchaya Kniga, Novosibirsk, 2002) [in Russian].Google Scholar
  9. 9.
    A. M. Vershik and V. Ya. Gershkovich, Itogi Nauki Tekh., Ser.: Sovr. Probl. Mat. Fundam. Napravleniya 16, 5–85 (1987).Google Scholar
  10. 10.
    Yu. L. Sachkov and E. F. Sachkova, Differ. Equations 53 (3), 352–365 (2017)CrossRefGoogle Scholar
  11. 11.
    A. A. Agrachev and Yu. L. Sachkov, Geometric Control Theory (Fizmatlit, Moscow, 2005) [in Russian].MATHGoogle Scholar
  12. 12.
    Yu. L. Sachkov, Sb.: Math. 194 (9), 1331–1359 (2003).MathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.Mechanics and Mathematics FacultyMoscow State UniversityMoscowRussia
  3. 3.Ailamazyan Program Systems InstituteRussian Academy of Sciencess. Ves’kovoRussia
  4. 4.RUDN UniversityMoscowRussia

Personalised recommendations