Advertisement

Regular and Chaotic Dynamics

, Volume 20, Issue 3, pp 383–400 | Cite as

The jacobi integral in nonholonomic mechanics

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

Abstract

In this paper we discuss conditions for the existence of the Jacobi integral (that generalizes energy) in systems with inhomogeneous and nonholonomic constraints. As an example, we consider in detail the problem of motion of the Chaplygin sleigh on a rotating plane and the motion of a dynamically symmetric ball on a uniformly rotating surface. In addition, we discuss illustrative mechanical models based on the motion of a homogeneous ball on a rotating table and on the Beltrami surface.

Keywords

nonholonomic constraint Jacobi integral Chaplygin sleigh rotating table Suslov problem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bizyaev, I. A., Nonintegrability and Obstructions to the Hamiltonianization of a Nonholonomic Chaplygin Top, Dokl. Math., 2014, vol. 90, no. 2, pp. 631–634; see also: Dokl. Akad. Nauk, 2014, vol. 458, no. 4, pp. 398–401.zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    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.zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Hamiltonization of Nonholonomic Systems in theNeighborhood of Invariant Manifolds, Regul. Chaotic Dyn., 2011, vol. 16, no. 5, pp. 443–464.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Rolling of a Ball without Spinning on a Plane: TheAbsence of an Invariant Measure in a System with a Complete Set of Integrals, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 571–579.zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Geometrisation of Chaplygin’s Reducing MultiplierTheorem, Nonlinearity, 2015 (in press).Google Scholar
  6. 6.
    Borisov, A. V., Kazakov, A.O., and Sataev, I. R., The Reversal and Chaotic Attractor in the NonholonomicModel of Chaplygin’s Top, Regul. Chaotic Dyn., 2014, vol. 19, no. 6, pp. 718–733.MathSciNetCrossRefGoogle Scholar
  7. 7.
    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.zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Hamiltonicity and Integrability of the Suslov Problem, Regul. Chaotic Dyn., 2011, vol. 16, nos. 1–2, pp. 104–116.zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Rolling of a Homogeneous Ball over a DynamicallyAsymmetric Sphere, Regul. Chaotic Dyn., 2011, vol. 16, no. 5, pp. 465–483.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., The Problem of Drift and Recurrence for the RollingChaplygin Ball, Regul. Chaotic Dyn., 2013, vol. 18, no. 6, pp. 832–859.zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    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
  12. 12.
    Borisov, A. V. and Mamaev, I. S., The Rolling Motion of a Rigid Body on a Plane and a Sphere: Hierarchyof Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 177–200.zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Borisov, A. V. and Mamaev, I. S., Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos, Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).Google Scholar
  14. 14.
    Borisov, A. V. and Mamaev, I. S., Conservation Laws, Hierarchy of Dynamics and Explicit Integrationof Nonholonomic Systems, Regul. Chaotic Dyn., 2008, vol. 13, no. 5, pp. 443–490.zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    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.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Borisov, A. V. and Mamaev, I. S., Topological Analysis of an Integrable System Related to the Rollingof a Ball on a Sphere, Regul. Chaotic Dyn., 2013, vol. 18, no. 4, pp. 356–371.zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Borisov, A. V. and Mamaev, I. S., The Dynamics of the Chaplygin Ball with a Fluid-Filled Cavity, Regul.Chaotic Dyn., 2013, vol. 18, no. 5, pp. 490–496; see also: Nelin. Dinam., 2012, vol. 8, no. 1, pp. 103–111.zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Borisov, A. V., Mamaev, I. S., and Bizyaev, I. A., The Hierarchy of Dynamics of a Rigid Body Rollingwithout Slipping and Spinning on a Plane and a Sphere, Regul. Chaotic Dyn., 2013, vol. 18, no. 3,pp. 277–328.zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Borisov, A. V., Mamaev, I. S., and Kilin, A.A., The Rolling Motion of a Ball on a Surface: New Integralsand Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 201–219.zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Borisov, A. V., Mamaev, I. S., and Kilin, A.A., Dynamics of Rolling Disk, Regul. Chaotic Dyn., 2003,vol. 8, no. 2, pp. 201–212.zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Bottema, O., On the Small Vibrations of Nonholonomic Systems, Indag. Math., 1949, vol. 11, pp. 296–298.Google Scholar
  22. 22.
    Burns, J. A., Ball Rolling on a Turntable: Analog for Charged Particle Dynamics, Amer. J. Phys., 1981,vol. 49, no. 1, pp. 56–58.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Carathéodory, C., Der Schlitten, Z. Angew. Math. Mech., 1933, vol. 13, no. 2, pp. 71–76.CrossRefGoogle Scholar
  24. 24.
    Earnshaw, S., Dynamics, or an Elementary Treatise on Motion, 3rd ed., Cambridge: Deighton, 1844.Google Scholar
  25. 25.
    Ehrlich, R. and Tuszynski, J., Ball on a Rotating Turntable: Comparison of theory and experiment, Amer. J. Phys., 1995, vol. 63, no. 4, pp. 351–359.CrossRefGoogle Scholar
  26. 26.
    Fassò, F. and Sansonetto, N., Conservation of ‘Moving’ Energy in Nonholonomic Systems with AffineConstraints and Integrability of Spheres on Rotating Surfaces, Preprint, arXiv:1503.06661 (2015).Google Scholar
  27. 27.
    Fassò, F. and Sansonetto, N., Conservation of Energy and Momenta in Nonholonomic Systems withAffine Constraints, Preprint, arXiv:1505.01172 (2015).Google Scholar
  28. 28.
    Fedorov, Yu.N. and Kozlov, V.V., Various Aspects of n-Dimensional Rigid Body Dynamics, Amer.Math. Soc. Transl. Ser.2, 1995, vol. 168, pp. 141–171.MathSciNetGoogle Scholar
  29. 29.
    Ferrario, C. and Passerini, A., Rolling Rigid Bodies and Forces of Constraint: An Application to AffineNonholonomic Systems, Meccanica, 2000, vol. 35, no. 5, pp. 433–442.zbMATHCrossRefGoogle Scholar
  30. 30.
    Fufaev, N.A., Rolling of a Heavy Homogeneous Ball over a Rough Sphere Rotating around a VerticalAxis, Soviet Appl. Mech., 1987, vol. 23, no. 1, pp. 86–89; see also: Prikl. Mekh., 1987, vol. 23, no. 1,pp. 98–101.zbMATHCrossRefGoogle Scholar
  31. 31.
    Fufaev, N.A., A Sphere Rolling on a Horizontal Rotating Plane, J. Appl. Math. Mech., 1983, vol. 47,no. 1, pp. 27–29; see also: Prikl. Mat. Mekh., 1983, vol. 47, no. 1, pp. 43–47.MathSciNetCrossRefGoogle Scholar
  32. 32.
    García-Naranjo, L.C., Maciejewski, A. J., Marrero, J.C., and Przybylska, M., The Inhomogeneous SuslovProblem, Phys. Lett. A, 2014, vol. 378, nos. 32–33, pp. 2389–2394.zbMATHCrossRefGoogle Scholar
  33. 33.
    Gersten, J., Soodak, H., and Tiersten, M. S., Ball Moving on Stationary or Rotating Horizontal Surface, Amer. J. Phys., 1992, vol. 60, no. 1, pp. 43–47.CrossRefGoogle Scholar
  34. 34.
    Hermans, J., A Symmetric Sphere Rolling on a Surface, Nonlinearity, 1995, vol. 8, no. 4, pp. 493–515.zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Karavaev, Yu. L. and Kilin, A.A., The Dynamics and Control of a Spherical Robot with an InternalOmniwheel Platform, Regul. Chaotic Dyn., 2015, vol. 20, no. 2, pp. 134–152.MathSciNetCrossRefGoogle Scholar
  36. 36.
    Kazakov, A. O., Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber BallRolling on a Plane, Regul. Chaotic Dyn., 2013, vol. 18, no. 5, pp. 508–520.zbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    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.zbMATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Kozlov, V.V., On the Theory of Integration of the Equations of Nonholonomic Mechanics, Regul. ChaoticDyn., 2002, vol. 7, no. 2, pp. 191–176.Google Scholar
  39. 39.
    Lynch, P. and Bustamante, M. D., Quaternion Solution for the Rock’n’Roller: Box Orbits, Loop Orbitsand Recession, Regul. Chaotic Dyn., 2013, vol. 18, nos. 1–2, pp. 166–183.zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Milne, E. A., Vectorial Mechanics, New York: Interscience, 1948.Google Scholar
  41. 41.
    Neimark, Ju. I. and Fufaev, N.A., Dynamics of Nonholonomic Systems, Trans. Math. Monogr., vol. 33,Providence, R.I.: AMS, 1972.Google Scholar
  42. 42.
    Pavlenko, Yu. G., Lectures on Theoretical Mechanics, Moscow: Fizmatlit, 2002 (Russian).Google Scholar
  43. 43.
    Rashevskii, P. K., A Course in Differential Geometry, 4th ed., Moscow: Editorial URSS, 2003 (Russian).Google Scholar
  44. 44.
    Routh, E. J., The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies: BeingPart II of a Treatise on the Whole Subject, 6th ed., New York: Dover, 1955.Google Scholar
  45. 45.
    Sokirko, A. V., Belopolskii, A. A., Matytsyn, A. V., and Kossakowski, D.A., Behavior of a Ballon the Surface of a Rotating Disk, Amer. J. Phys., 1994, vol. 62, no. 2, pp. 151–156.CrossRefGoogle Scholar
  46. 46.
    Soodak, H. and Tiersten, M. S., Perturbation Analysis of Rolling Friction on a Turntable, Amer. J. Phys., 1996, vol. 64, no. 9, pp. 1130–1139.CrossRefGoogle Scholar
  47. 47.
    Tokieda, T., Roll Models, Amer. Math. Monthly, 2013, vol. 120, no. 3, pp. 265–282.zbMATHMathSciNetCrossRefGoogle Scholar
  48. 48.
    Vierkandt, A., Über gleitende und rollende Bewegung, Monatsh. Math. Phys., 1892, vol. 3, no. 1, pp. 31–38, 97–116.zbMATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    Weckesser, W., A Ball Rolling on a Freely Spinning Turntable, Amer. J. Phys., 1997, vol. 65, no. 8,pp. 736–738.CrossRefGoogle Scholar
  50. 50.
    Weltner, K., Stable Circular Orbits of Freely Moving Balls on Rotating Discs, Amer. J. Phys., 1979,vol. 47, no. 11, pp. 984–986.CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  • Alexey V. Borisov
    • 1
    • 2
  • Ivan. S. Mamaev
    • 2
    • 3
  • Ivan. A. Bizyaev
    • 2
    • 4
  1. 1.Moscow Institute of Physics and TechnologyDolgoprudnyiRussia
  2. 2.Udmurt State UniversityIzhevskRussia
  3. 3.M. T. Kalashnikov Izhevsk State Technical UniversityIzhevskRussia
  4. 4.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

Personalised recommendations