Regular and Chaotic Dynamics

, Volume 20, Issue 4, pp 401–427 | Cite as

The dynamics of systems with servoconstraints. II



This paper addresses the dynamics of systems with servoconstraints where the constraints are realized by controlling the inertial properties of the system. Vakonomic systems are a particular case. Special attention is given to the motion on Lie groups with left-invariant kinetic energy and a left-invariant constraint. The presence of symmetries allows the dynamical equations to be reduced to a closed system of differential equations with quadratic right-hand sides. As the main example, we consider the rotation of a rigid body with a left-invariant servo-constraint, which implies that the projection of the body’s angular velocity on some body-fixed direction is zero.


servoconstraints symmetries Lie groups left-invariant constraints systems with quadratic right-hand sides vakonomic systems 

MSC2010 numbers

70E18 34C40 


  1. 1.
    Kozlov, V.V., Principles of Dynamics and Servoconstraints, Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1989, no. 5, pp. 59–66 (Russian).Google Scholar
  2. 2.
    Kozlov, V.V., The Dynamics of Systems with Servoconstaints: 1, Regul. Chaotic Dyn., 2015, vol. 20, no. 3, pp. 205–224.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Kozlov, V.V., Dynamics of Systems with Nonintegrable Constraints: 1, Mosc. Univ. Mech. Bull., 1982, vol. 37, nos. 3–4, pp. 27–34; see also: Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1982, no. 3, pp. 92–100; Dynamics of Systems with Nonintegrable Constraints: 2, Mosc. Univ. Mech. Bull., 1982, vol. 37, nos. 3–4, pp. 74–80; see also: Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1982, no. 4, pp. 70–76; Dynamics of Systems with Nonintegrable Constraints: 3, Mosc. Univ. Mech. Bull., 1983, vol. 38, no. 3, pp. 40–51; see also: Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1983, no. 3, pp. 102–111; Dynamics of Systems with Nonintegrable Constraints: 4. Integral Principles, Mosc. Univ. Mech. Bull., 1987, vol. 42, no. 5, pp. 40–49; see also: Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1987, no. 5, pp. 76–83; Dynamics of Systems with Nonintegrable Constraints: 5. Freedom Principle and Ideal Constraints Condition, Mosc. Univ. Mech. Bull., 1988, vol. 43, no. 6, pp. 23–29; see also: Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1988, no. 6, pp. 51–54.MATHGoogle Scholar
  4. 4.
    Li, Sh.-M. and Berakdar, J., On the Validity of the Vakonomic Model and the Chetaev Model for Constraint Dynamical Systems, Rep. Math. Phys., 2007, vol. 60, no. 1, pp. 107–116.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Béghin, M. H., Étude théorique des compas gyrostatiques Anschütz et Sperry, Paris: Impr. nationale, 1921; see also: Annales hydrographiques, 1921.Google Scholar
  6. 6.
    Appel, P., Traité de Mécanique rationnelle: Vol. 2. Dynamique des systèmes. Mécanique analytique, 6th ed. Paris: Gauthier-Villars, 1953.Google Scholar
  7. 7.
    Arnol’d, V. I., Kozlov, V.V., and Neïshtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, 3rd ed., Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 2006.Google Scholar
  8. 8.
    Kozlov, V.V., Realization of Nonintegrable Constraints in Classical Mechanics, Sov. Phys. Dokl., 1983, vol. 28, pp. 735–737; see also: Dokl. Akad. Nauk SSSR, 1983, vol. 272, no. 3, pp. 550–554.MATHGoogle Scholar
  9. 9.
    Kozlov, V.V., On the Realization of Constraints in Dynamics, J. Appl. Math. Mech., 1992, vol. 56, no. 4, pp. 594–600; see also: Prikl. Mat. Mekh., 1992, vol. 56, no. 4, pp. 692–698.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Volterra, V., Sopra una classe di equazioni dinamiche, Atti della R. Accad. Sci. di Torino, 1898, vol. 33, pp. 471–475.Google Scholar
  11. 11.
    Poincaré, H., Sur une forme nouvelle des équations de la Mécanique, C. R. Acad. Sci., 1901, vol. 132, pp. 369–371.MATHGoogle Scholar
  12. 12.
    Kozlov, V.V., General Theory of Vortices, Encyclopaedia Math. Sci., vol. 67, Berlin: Springer, 2003.MATHGoogle Scholar
  13. 13.
    Borisov, A.V. and Mamaev, I. S., Modern Methods of the Theory of Integrable Systems, Moscow: R&C Dynamics, ICS, 2003 (Russian).MATHGoogle Scholar
  14. 14.
    Kozlov, V.V., On the Existence of an Integral Invariant of a Smooth Dynamic System, J. Appl. Math. Mech., 1987, vol. 51, no. 4, pp. 420–426; see also: Prikl. Mat. Mekh., 1987, vol. 51, no. 4, pp. 538–545.MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Volterra, V., Sur la théorie des variations des latitudes, Acta Math., 1899, vol. 22, pp. 201–357.MathSciNetCrossRefGoogle Scholar
  16. 16.
    de La Vallée Poussin, Ch.-J., Cours d’analyse infinitesimale: Vol. 2, 2nd ed., Paris: Gauthier-Villars, 1912.Google Scholar
  17. 17.
    Nambu, Y., Generalized Hamiltonian Dynamics, Phys. Rev. D(3), 1973, vol. 7, no. 8, pp. 2405–2412.MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kozlov, V.V., Dynamical Systems on a Torus with Multivalued Integrals, Proc. Steklov Inst. Math., 2007, vol. 256, no. 1, pp. 188–205; see also: Tr. Mat. Inst. Steklova, 2007, vol. 256, pp. 201–218.MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Yoshida, H., Necessary Condition for the Existence of Algebraic First Integrals, Celestial Mechanics, 1983, vol. 31, no. 4, pp. 363–399.MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kozlov, V.V., Tensor Invariants of Quasihomogeneous Systems of Differential Equations, and the Asymptotic Kovalevskaya — Lyapunov Method, Math. Notes, 1992, vol. 51, nos. 1–2, pp. 138–142; see also: Mat. Zametki, 1992, vol. 51, no. 2, pp. 46–52.MathSciNetCrossRefGoogle Scholar
  21. 21.
    Darboux, G., Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal: Vol. 1, Paris: Gauthier-Villars, 1887.Google Scholar
  22. 22.
    Kozlov, V.V., Remarks on Integrable Systems, Regul. Chaotic Dyn., 2014, vol. 19, no. 2, pp. 145–161.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Jacobi, C.G. J., Sur la rotation d’un corps, in Gesammelte Werke: Vol. 2, Berlin: Raimer, 1882, pp. 289–352.Google Scholar
  24. 24.
    Kozlov, V.V., Methods of Qualitative Analysis in the Dynamics of a Rigid Body, 2nd ed., Moscow-Izhevsk: R&C Dynamics, Institute of Computer Science, 2000. (Russian).MATHGoogle Scholar
  25. 25.
    Naimark, M.A., Linear Representations of the Lorentz Group, Internat. Ser. Monogr. Pure Appl. Math., vol. 63, Oxford: Pergamon, 1964.Google Scholar
  26. 26.
    Kozlov, V.V., Invariant Measures of the Euler -Poincaré Equations on Lie Algebras, Funct. Anal. Appl., 1988, vol. 22, no. 1, pp. 58–59; see also: Funktsional. Anal. i Prilozhen., 1988, vol. 22, no. 1, pp. 69–70.MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Kozlov, V.V. and Yaroshchuk, V.A., On Invariant Measures of Euler — Poincaré Equations on Unimodular Groups, Mosc. Univ. Mech. Bull., 1993, vol. 48, no. 2, pp. 45–50; see also: Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., 1993, no. 2, pp. 91–95.MathSciNetMATHGoogle Scholar
  28. 28.
    Veselov, A.P. and Veselova, L. E., Flows on Lie Groups with a Nonholonomic Constraint and Integrable Non-Hamiltonian Systems, Funct. Anal. Appl., 1986, vol. 20, no. 5, pp. 308–309; see also: Funktsional. Anal. i Prilozhen., 1986, vol. 20, no. 4, pp. 65–66.MathSciNetMATHGoogle Scholar
  29. 29.
    Lyapunov, A. M., On Constant Screw Motions of a Rigid Body in a Liquid, Soobshch. Kharkov. Mat. Obshch., Ser. 2, 1888, vol. 1, nos. 1–2, pp. 7–60 (Russian).Google Scholar
  30. 30.
    Borisov, A.V. and Mamaev, I. S., Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos, Moscow-Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).Google Scholar
  31. 31.
    Zenkov, D.V. and Kozlov, V.V., Poinsot’s Geometric Representation in the Dynamics of a Multidimensional Rigid Body, Trudy Sem. Vektor. Tenzor. Anal., 1988, vol. 23, pp. 202–204 (Russian).MathSciNetMATHGoogle Scholar
  32. 32.
    Fedorov, Yu.N. and Kozlov, V.V., Various Aspects of n-Dimensional Rigid Body Dynamics, in Dynamical Systems in Classical Mechanics, Amer. Math. Soc. Transl. Ser. 2, vol. 168, Providence, R.I.: AMS, 1995, pp. 141–171.Google Scholar
  33. 33.
    Vershik, A.M. and Gershkovich, V.Ya., Nonholonomic Dynamical Systems, Geometry of Distributions and Variational Problems, in Dynamical Systems: VII. Integrable Systems Nonholonomic Dynamical Systems, V. I. Arnol’d, S.P. Novikov (Eds.), Encyclopaedia Math. Sci., vol. 16, Berlin: Springer, 1994.Google Scholar
  34. 34.
    Montgomery, R., A Tour of Subriemannian Geometries, Their Geodesics and Applications, Math. Surveys Monogr., vol. 91, Providence, R.I.: AMS, 2006.CrossRefGoogle Scholar
  35. 35.
    Lewis, A.D. and Murray, R.M., Variational Principles for Constrained Systems: Theory and Experiment, Internat. J. Non-Linear Mech., 1995, vol 30, no. 6, pp. 793–815.MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Favretti, M., Equivalence of Dynamics for Nonholonomic Systems with Transverse Constraints, J. Dynam. Differential Equations, 1998, vol. 10, no. 4, pp. 511–536.MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Cardin, F. and Favretti, M., On Nonholonomic and Vakonomic Dynamics of Mechanical Systems with Nonintegrable Constraints, J. Geom. Phys., 1996, vol. 18, no. 4, pp. 295–325.MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    de Leon, M., Marrero, J.C., and Martin de Diego, D., Vakonomic Mechanics versus Non-Holonomic Mechanics: A Unified Geometrical Approach, J. Geom. Phys., 2000, vol. 35, nos. 2–3, pp. 126–144.MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Zampieri, G., Nonholonomic versus Vakonomic Dynamics, J. Differential Equations, 2000, vol. 163, no. 2, pp. 335–347.MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Fernandez, O.E. and Bloch, A.M., Equivalence of the Dynamics of Nonholonomic and Variational Nonholonomic Systems for Certain Initial Data, J. Phys. A, 2008, vol. 41, no. 34, 344005, 20 pp.Google Scholar
  41. 41.
    Deryabin, M.V. and Kozlov, V.V., On Effect of “emerging” of a Heavy Rigid Body in a Fluid, Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, 2002, no. 1, pp. 68–74 (Russian). 68-74.Google Scholar
  42. 42.
    Antunes, A. C. B. and Sigaud, C., Controlling Nonholonomic Chaplygin Systems, Braz. J. Phys., 2010, vol. 40, no. 2, pp. 131–140.CrossRefGoogle Scholar
  43. 43.
    Bolotin, S.V., The Problem of Optimal Control of a Chaplygin Ball by Inernal Rotors, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 559–570.MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Llibre, J., Ramirez, R., and Sadovskaia, N., A New Approach to the Vakonomic Mechanics, Nonlinear Dynam., 2014, vol. 78, no. 3, pp. 2219–2247.MathSciNetCrossRefGoogle Scholar
  45. 45.
    Gledzer, E.B., Dolzhanskii, F. V., and Obukhov, A. M., Systems of Hydrodynamical Type and Their Applications, Moscow: Nauka, 1981 (Russian).Google Scholar
  46. 46.
    Borisov, A.V., Mamaev, I. S., and Tsyganov, A.V., Nonholonomic Dynamics and Poisson Geometry, Russian Math. Surveys, 2014, vol. 69, no. 3, pp. 481–538; see also: Uspekhi Mat. Nauk, 2014, vol. 69, no. 3(417), pp. 87–144.MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Dynamics and Control of an Omniwheel Vehicle, Regul. Chaotic Dyn., 2015, vol. 20, no. 2, pp. 153–172.MathSciNetCrossRefGoogle Scholar
  48. 48.
    Patera, J., Sharp, R.T., and Winternitz, P., Invariants of Real Low Dimension Lie Algebras, J. Math. Phys., 1976, vol. 17, no. 6, pp. 986–994.MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Cariñena, J. F., Ibort, A., Marmo, G., and Perelomov, A., On the Geometry of Lie Algebras and Poisson Tensors, J. Phys. A, 1994, vol. 27, no. 22, pp. 7425–7449.MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Hamel, G., Das Hamiltonsche Prinzip bei nichtholonomen Systemen, Math. Ann., 1935, vol. 111, no. 1, pp. 94–97.MathSciNetCrossRefGoogle Scholar
  51. 51.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., How To Control Chaplygin’s Sphere Using Rotors, Regul. Chaotic Dyn., 2012, vol. 17, nos. 3–4, pp. 258–272.MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., How To Control Chaplygin’s Sphere Using Rotors: 2, Regul. Chaotic Dyn., 2013, vol. 18, nos. 1–2, pp. 144–158.MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Svinin, M., Morinaga, A., and Yamamoto, M., On the Dynamic Model and Motion Planning for a Spherical Rolling Robot Actuated by Orthogonal Internal Rotors, Regul. Chaotic Dyn., 2013, vol. 18, nos. 1–2, pp. 126–143.MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Geometrisation of Chaplygin’s Reducing Multiplier Theorem, Nonlinearity, 2015, vol. 28, no. 7, pp. 2307–2318.CrossRefGoogle Scholar
  55. 55.
    Borisov, A. V. and Mamaev, I. S., Topological Analysis of an Integrable System Related to the Rolling of a Ball on a Sphere, Regul. Chaotic Dyn., 2013, vol. 18, no. 4, pp. 356–371.MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., The Problem of Drift and Recurrence for the Rolling Chaplygin Ball, Regul. Chaotic Dyn., 2013, vol. 18, no. 6, pp. 832–859.MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Bizyaev, I.A., Borisov, A.V., and Mamaev, I. S., The Dynamics of Nonholonomic Systems Consisting of a Spherical Shell with a Moving Rigid Body Inside, Regul. Chaotic Dyn., 2014, vol. 19, no. 2, pp. 198–213.MathSciNetCrossRefGoogle Scholar
  58. 58.
    Borisov, A. V., Mamaev, I. S., and Bizyaev, I. A., The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere, Regul. Chaotic Dyn., 2013, vol. 18, no. 3, pp. 277–328.MathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    Marle, C. M., Kinematic and Geometric Constraints, Servomechanisms and Control of Mechanical Systems, Rend. Sem. Mat. Univ. Politec. Torino, 1996, vol. 54, no. 4, pp. 353–364.MathSciNetMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

Personalised recommendations