Regular and Chaotic Dynamics

, Volume 21, Issue 1, pp 136–146 | Cite as

Dynamics of the Chaplygin sleigh on a cylinder

  • Ivan A. Bizyaev
  • Alexey V. Borisov
  • Ivan S. Mamaev


This paper is concerned with the motion of the Chaplygin sleigh on the surface of a circular cylinder. In the case of inertial motion, the problem reduces to the study of the dynamical system on a (two-dimensional) torus and to the classification of singular points. Particular cases in which the system admits an invariant measure are found.

In the case of a balanced and dynamically symmetric Chaplygin sleigh moving in a gravitational field it is shown that on the average the system has no drift along the vertical.


Chaplygin sleigh invariant measure nonholonomic mechanics 

MSC2010 numbers

37J60 37C10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    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.CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Brill, A., Vorlesungen zur Einführung in die Mechanik raumerfüllender Massen, Leipzig: Teubner, 1909.zbMATHGoogle Scholar
  3. 3.
    Carathéodory, C., Der Schlitten, Z. Angew. Math. Mech., 1933, vol. 13, no. 2, pp. 71–76.CrossRefGoogle Scholar
  4. 4.
    Borisov, A. V. and Mamaev, I. S., The Dynamics of a Chaplygin Sleigh, J. Appl. Math. Mech., 2009, vol. 73, no. 2, pp. 156–161; see also: Prikl. Mat. Mekh., 2009, vol. 73, no. 2, pp. 219–225.CrossRefMathSciNetGoogle Scholar
  5. 5.
    Chaplygin, S.A., On the Theory ofMotion of Nonholonomic Systems. The Reducing-Multiplier Theorem, Regul. Chaotic Dyn., 2008, vol. 13, no. 4, pp. 369–376; see also: Mat. Sb., 1912, vol. 28, no. 2, pp. 303–314.CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    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.CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    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.CrossRefMathSciNetGoogle Scholar
  8. 8.
    Bolotin, S. V. and Popova, T. V., On the Motion of a Mechanical System inside a Rolling Ball, Regul. Chaotic Dyn., 2013, vol. 18, nos. 1–2, pp. 159–165.CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    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.CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Chaplygin, S. A., On a Ball’s Rolling on a Horizontal Plane, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 131–148; see also: Math. Sb., 1903, vol. 24, no. 1, pp. 139–168.CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    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.CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Oreshkina, L.N., Some Generalizations of the Chaplygin Sleigh Problem, Mekh. Tverd. Tela, 1986, no. 19, pp. 34–39 (Russian).MathSciNetGoogle Scholar
  13. 13.
    Ifraimov, S.V. and Kuleshov, A. S., On Moving Chaplygin Sleigh on a Convex Surface, Autom. Remote Control, 2013, vol. 74, no. 8, pp. 1297–1306; see also: Avtomat. i Telemekh., 2013, no. 8, pp. 80–90.CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., On a Nonholonomic Dynamical Problem, Math. Notes, 2006, vol. 79, nos. 5–6, pp. 734–740; see also: Mat. Zametki, 2006, vol. 79, no. 5, pp. 790–796.CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Noohi, E., Mahdavi, S. S., Baghani, A., and Ahmadabadi, M.N., Wheel-Based Climbing Robot: Modeling and Control, Advanced Robotics, 2010, vol. 24, nos. 8–9, pp. 1313–1343.CrossRefGoogle Scholar
  16. 16.
    Hamel, G., Die Lagrange–Eulerschen Gleichungen der Mechanik, Z. Math. u. Phys., 1904, vol. 50, pp. 1–57.zbMATHGoogle Scholar
  17. 17.
    Kozlov, V.V., The Euler–Jacobi–Lie Integrability Theorem, Regul. Chaotic Dyn., 2013, vol. 18, no. 4, pp. 329–343.CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Borisov, A.V. and Mamaev, I. S., Strange Attractors in Rattleback Dynamics, Physics–Uspekhi, 2003, vol. 46, no. 4, pp. 393–403; see also: Uspekhi Fiz. Nauk, 2003, vol. 173, no. 4, pp. 407–418.MathSciNetGoogle Scholar
  19. 19.
    Borisov, A.V., Kazakov, A.O., and Kuznetsov, S.P., Nonlinear Dynamics of the Rattleback: A Nonholonomic Model, Physics–Uspekhi, 2014, vol. 57, no. 5, pp. 453–460; see also: Uspekhi Fiz. Nauk, 2014, vol. 184, no. 5, pp. 493–500.Google Scholar
  20. 20.
    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.CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Mushtari, Kh. M., über das Abrollen eines schweren starren Rotationskörpers auf einer unbeweglichen horizontalen Ebene, Mat. Sb., 1932, vol. 39, nos. 1–2, pp. 105–126 (Russian).Google Scholar
  22. 22.
    Bobylev, D., Kugel, die ein Gyroskop einschliesst und auf einer Horizontalebene rollt, ohne dabei zu gleiten, Mat. Sb., 1892, vol. 16, no. 3, pp. 544–581 (Russian).Google Scholar
  23. 23.
    Nekrassov, P.A., étude analytique d’un cas du mouvement d’un corps pesant autour d’un point fixe, Mat. Sb., 1896, vol. 18, no. 2, pp. 161–274 (Russian).Google Scholar
  24. 24.
    Nekrassov, P.A., Zur Frage von der Bewegung eines schweren starren Körpers um einen festen Punkt, Mat. Sb., 1892, vol. 16, no. 3, pp. 508–517 (Russian).Google Scholar
  25. 25.
    Goriatchev, D.N., Sur le mouvement d’un solide pesant autour d’un point fixe dans le cas A = B = 4C, Mat. Sb., 1900, vol. 21, no. 3, pp. 431–438 (Russian).Google Scholar
  26. 26.
    Appelroth, H. H., Sur les cas particuliers les plus simples du mouvement d’un gyroscope pesant asymmétrique de M-me Kowalewsky, Mat. Sb., 1910, vol. 27, no. 3, pp. 262–334 (Russian).Google Scholar
  27. 27.
    Appelroth, H. H., Sur les cas particuliers les plus simples du mouvement d’un gyroscope pesant asymmétrique de M-me Kowalewsky (2-me article), Mat. Sb., 1911, vol. 27, no. 4, pp. 477–559 (Russian).Google Scholar
  28. 28.
    Sloudsky, Th., Note relative au problème de plusieurs corps, Mat. Sb., 1879, vol. 9, no. 3, pp. 536–545 (Russian).Google Scholar
  29. 29.
    Kakehashi, Y., Izawa, T., Shirai, T., Nakanishi, Y., Okada, K., and Inaba, M., Achievement of Hula Hooping by Robots through Deriving Principle Structure Towards Flexible Spinal Motion, J. Robot. Mechatron., 2012, vol. 24, no. 3, pp. 540–546.Google Scholar
  30. 30.
    Caughey, T. K., Hula-Hoop: An Example of Heteroparametric Excitation, Amer. J. Phys., 1960, vol. 28, no. 2, pp. 104–109.CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    Fedorov, Yu.N., García-Naranjo, L.C., and Marrero, J.C., Unimodularity and Preservation of Volumes in Nonholonomic Mechanics, J. Nonlinear Sci., 2015, vol. 25, no. 1, pp. 203–246.CrossRefMathSciNetzbMATHGoogle Scholar
  32. 32.
    Kozlov, V.V., Several Problems on Dynamical Systems and Mechanics, Nonlinearity, 2008, vol. 21, no. 9, T149–T155.CrossRefzbMATHGoogle Scholar
  33. 33.
    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.CrossRefMathSciNetzbMATHGoogle Scholar
  34. 34.
    Bilimovitch, A.D., La pendule nonholonome, Mat. Sb., 1914, vol. 29, no. 2, pp. 234–240 (Russian).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  • Ivan A. Bizyaev
    • 1
  • Alexey V. Borisov
    • 1
  • Ivan S. Mamaev
    • 1
  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

Personalised recommendations