Differential Equations

, Volume 51, Issue 11, pp 1476–1483 | Cite as

Superintegrability of left-invariant sub-Riemannian structures on unimodular three-dimensional Lie groups

Control Theory


We consider left-invariant sub-Riemannian problems on three-dimensional unimodular Lie groups. We show that the Hamiltonian system of the Pontryagin maximum principle for such problems is Liouville integrable and even superintegrable (i.e., has four independent integrals, three of which are in involution).


Hamiltonian System Hamiltonian Vector Pontryagin Maximum Principle Casimir Function Liouville Integrability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Sachkov, Yu.L., Control Theory on Lie Groups, J. Math. Sci., 2009, vol. 156, no. 3, pp. 381–439.CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Brockett, R., Control Theory and Singular Riemannian Geometry, in New Direction in Applied Mathematics, New York, 1981, pp. 11–27.Google Scholar
  3. 3.
    Vershik, A.M. and Gershkovich, V.Ya., Nonholonomic Dynamical Systems. Geometry of Distributions and Variational Problems, Dinam. Sist.–7: Itogi Nauki i Tekhn. Sovr. Probl. Mat. Fund. Napravl., 1987, vol. 16, pp. 5–85.MathSciNetGoogle Scholar
  4. 4.
    Boscain, U. and Rossi, F., Invariant Carnot–CaratheodoryMetrics on S3, SO(3), SL(2) and Lens Spaces, SIAM J. Control Optim., 2008, vol. 47, pp. 1851–1878.CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Sachkov, Yu.L., Cut Locus and Optimal Synthesis in the Sub-Riemannian Problem on the Group of Motions of a Plane, ESAIM Control Optim. Calc. Var., 2011, vol. 17, pp. 293–321.CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Ardentov, A.A. and Sachkov, Yu.L., Conjugate Points in Nilpotent Sub-Riemannian Problem on the Engel Group, J. Math. Sci., 2013, vol. 195, no. 3, pp. 369–390.CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Barilari, D., Boscain, U., and Gauthier, J.P., On 2-Step, Corank 2 Nilpotent Sub-Riemannian Metrics, SIAM J. Control Optim., 2011, vol. 50, pp. 559–582.CrossRefMathSciNetGoogle Scholar
  8. 8.
    Montgomery, R., Shapiro, M., and Stolin, A., A Nonintegrable Sub-Riemannian Geodesic Flow on a Carnot Group, J. Dynam. Control Systems, 1997, vol. 3, pp. 519–530.MathSciNetMATHGoogle Scholar
  9. 9.
    Mashtakov, A.P. and Sachkov, Yu.L., Integrability of Left-Invariant Sub-Riemannian Structures on the Special Linear Group SL2(R), Differ. Uravn., 2014, vol. 50, no. 11, pp. 1541–1547.CrossRefMathSciNetGoogle Scholar
  10. 10.
    Jurdjevic, V., Integrable Hamiltonian Systems on Complex Lie Groups, Mem. Amer. Math. Soc., 2005, vol. 178, no. 838.Google Scholar
  11. 11.
    Arnol’d, V.I., Matematicheskie metody klassicheskoi mekhaniki (Mathematical Methods of Classical Mechanics), Moscow: Nauka, 1989.Google Scholar
  12. 12.
    Nekhoroshev, N.N., Action-Angle Variables and Their Generalizations, Trans. Moscow Math. Soc., 1972, vol. 26, pp. 180–198.Google Scholar
  13. 13.
    Mishchenko, A.S. and Fomenko, A.T., Generalized Liouville Method for the Integration of Hamiltonian Systems, Funktsional Anal. i Prilozhen., 1978, vol. 12, no. 2, pp. 46–56.CrossRefMATHGoogle Scholar
  14. 14.
    Fasso, F., Superintegrable Hamiltonian Systems: Geometry and Applications, Acta Appl. Math., 2005, vol. 87, pp. 93–121.CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Agrachev, A.A. and Barilari, D., Sub-Riemannian Structures on 3D Lie Groups, J. Dynam. Control Systems, 2012, vol. 18, pp. 21–41.CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Kirillov, A.A., Lectures on the Orbit Method, Grad. Stud. Math. 64, New York, 2004.Google Scholar
  17. 17.
    Belläiche, A., The Tangent Space in Sub-Riemannian Geometry, Sub-Riemannian Geometry, 1996, vol. 144, pp. 1–78.CrossRefGoogle Scholar
  18. 18.
    Jean, F., Sub-Riemannian Geometry, in Lectures Given at the Trimester on Dynamical and Control Systems, Trieste, September–December. 2003.Google Scholar
  19. 19.
    Butt, Ya., Sachkov, Yu., and Bhatti, A., Extremal Trajectories and Maxwell Strata in Sub-Riemannian Problem on Group of Motions of Pseudo Euclidean Plane, J. Dynam. Control Systems, 2014, vol. 20, no. 3, pp. 341–364.CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Mazhitova, A.D., Sub-Riemannian Geodesics on the Three-Dimensional Solvable Non-Nilpotent Lie Group SOLV-, J. Dynam. Control Systems, 2012, vol. 18, no. 3, pp. 309–322.CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Bonnard, B., Cots, O., and Shcherbakova, N., The Serret–Andoyer Riemannian Metric and Euler–Poinsot Rigid Body Motion, Math. Control Related Fields, 2013, vol. 3, pp. 287–302.CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Eindhoven University of TechnologyEindhovenNetherlands
  2. 2.University of Hradec KrálovéHradec KrálovéCzech Republic

Personalised recommendations