Regular and Chaotic Dynamics

, Volume 20, Issue 5, pp 553–604 | Cite as

Symmetries and reduction in nonholonomic mechanics

Article

Abstract

This paper is a review of the problem of the constructive reduction of nonholonomic systems with symmetries. The connection of reduction with the presence of the simplest tensor invariants (first integrals and symmetry fields) is shown. All theoretical constructions are illustrated by examples encountered in applications. In addition, the paper contains a short historical and critical sketch covering the contribution of various researchers to this problem.

Keywords

reduction symmetry tensor invariant first integral symmetry group symmetry field nonholonomic constraint Noether theorem 

MSC2010 numbers

37J60 37J35 37C10 

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References

  1. 1.
    Albouy, A., On the Force Fields Which Are Homogeneous of Degree -3, arXiv:1412.4150 (2014).Google Scholar
  2. 2.
    Albouy, A., There Is a Projective Dynamics, Eur. Math. Soc. Newsl., 2013, vol. 89, pp. 37–43.MATHMathSciNetGoogle Scholar
  3. 3.
    Albouy, A. and Chenciner, A., Le problème des n corps et les distances mutuelles, Invent. Math., 1998, vol. 131, no. 1, pp. 151–184.MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Aref, H. and Stremler, M.A., On the Motion of Three Point Vortices in a Periodic Strip, J. Fluid Mech., 1996, vol. 314, pp. 1–25.MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Arnol’d, V. I., Geometrical Methods in the Theory of Ordinary Differential Equations, Grundlehren Math. Wiss., vol. 250, New York: Springer, 2012.Google Scholar
  6. 6.
    Arnol’d, V. I., Kozlov, V.V., and Neishtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, 3rd ed., Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 2006.Google Scholar
  7. 7.
    Balseiro, P. and Fernandez, O.E., Reduction of Nonholonomic Systems in Two Stages, arXiv:1409.0456 (2014).Google Scholar
  8. 8.
    Birkhoff, G.D., Dynamical Systems with Two Degrees of Freedom, Trans. Amer. Math. Soc., 1917, vol. 18, no. 2, pp. 199–300.MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Bizyaev, I. A. and Tsiganov, A.V., On the Routh Sphere Problem, J. Phys. A, 2013. vol. 46, no. 8, 085202, 11 pp.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bloch, A., Nonholonomic Mechanics and Control, Interdiscip. Appl. Math., vol. 24, New York: Springer, 2003.Google Scholar
  11. 11.
    Bloch, A. M., Krishnaprasad, P. S., Marsden, J.E., Murray, R. M., Nonholonomic Mechanical Systems with Symmetry, Arch. Rational Mech. Anal., 1996, vol. 136, no. 1, pp. 21–99.MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Hamiltonization of Non-Holonomic Systems in the Neighborhood of Invariant Manifolds, Regul. Chaotic Dyn., 2011, vol. 116, no. 5, pp. 443–464.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Lie Algebras in Vortex Dynamics and Celestial Mechanics: 4, Regul. Chaotic Dyn., 1999, vol. 4, no. 1, pp. 23–50.MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Topology and Stability of Integrable Systems, Russian Math. Surveys, 2010, vol. 65, no. 2, pp. 259–318; see also: Uspekhi Mat. Nauk, 2010, vol. 65, no. 2, pp. 71–132.MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    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.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Multiparticle Systems: The Algebra of Integrals and Integrable Cases, Regul. Chaotic Dyn., 2009, vol. 14, no. 1, pp. 18–41.MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    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.MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Borisov, A. V., Lutsenko, S. G., and Mamaev, I. S., Dynamics of a Wheeled Carriage on a Plane, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2010, no. 4, pp. 39–48 (Russian).Google Scholar
  19. 19.
    Borisov, A.V. and Mamaev, I. S., An Integrable System with a Nonintegrable Constraint, Math. Notes, 2006, vol. 80, nos. 1–2, pp. 127–130; see also: Mat. Zametki, 2006, vol. 80, no. 1, pp. 131–134.MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Borisov, A. V. and Mamaev, I. S., Chaplygin’s Ball Rolling Problem Is Hamiltonian, Math. Notes, 2001, vol. 70, no. 5, pp. 720–723; see also: Mat. Zametki, 2001, vol. 70, no. 5, pp. 793–795.MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Borisov, A.V. and Mamaev, I. S., Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems, Regul. Chaotic Dyn., 2008, vol. 13, no. 5, pp. 443–490.MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Borisov A.V., Mamaev I. S. Dynamics of a Rigid Body: Hamiltonian Methods, Integrability, Chaos, 2nd ed., Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).Google Scholar
  23. 23.
    Borisov, A.V. and Mamaev, I. S., Modern Methods of the Theory of Integrable Systems, Izhevsk: Institute of Computer Science, 2003 (Russian).MATHGoogle Scholar
  24. 24.
    Borisov, A.V. and Mamaev, I. S., Non-Linear Poisson Brackets and Isomorphisms in Dynamics, Regul. Chaotic Dyn., 1997, vol. 2, nos. 3–4, pp. 72–89 (Russian).MATHMathSciNetGoogle Scholar
  25. 25.
    Borisov, A.V. and Mamaev, I. S., Reduction in the Two-Body Problem on the Lobatchevsky Plane, Nelin. Dinam., 2006, vol. 2, no. 3, pp. 279–285 (Russian).MathSciNetGoogle Scholar
  26. 26.
    Borisov, A. V. and Mamaev, I. S., Rolling of a Rigid Body on a Plane and Sphere: Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 177–200.MATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    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
  28. 28.
    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
  29. 29.
    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.MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Borisov, A. V., Mamaev, I. S., and Bizyaev, I. A., The Jacobi Integral in Nonholonomic Mechanics, Regul. Chaotic Dyn., 2015, vol. 20, no. 3, pp. 383–400.MathSciNetCrossRefGoogle Scholar
  31. 31.
    Borisov, A. V., Mamaev, I. S., and Kilin, A.A., The Rolling Motion of a Ball on a Surface: New Integrals and Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 201–219.MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Borisov, A. V., Mamaev, I. S., and Kilin, A.A., Two-Body Problem on a Sphere: Reduction, Stochasticity, Periodic Orbits, Regul. Chaotic Dyn., 2004, vol. 9, no. 3, pp. 265–279.MATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Borisov, A. V. and Pavlov, A.E., Dynamics and Statics of Vortices on a Plane and a Sphere: 1, Regul. Chaotic Dyn., 1998, vol. 3, no. 1, pp. 28–38.MATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Bottema, O., Die Bewegung eines einfachenWagenmodells, Z. Angew. Math. Mech., 1964, vol. 44, no. 12, pp. 585–593.MATHCrossRefGoogle Scholar
  35. 35.
    Broer, H. and Simó, C., Hill’s Equation with Quasi-Periodic Forcing: Resonance Tongues, Instability Pockets and Global Phenomena, Bol. Soc. Brasil. Mat. (N. S.), 1998, vol. 29, no. 2, pp. 253–293.MATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Cartan, é., Lessons on Integral Invariants, Paris: Hermann, 1922.Google Scholar
  37. 37.
    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.MATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    de Le´on, M., A Historical Review on Nonholomic Mechanics, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 2012, vol. 106, no. 1, pp. 191–224.CrossRefGoogle Scholar
  39. 39.
    Dullin, H.R., The Lie–Poisson Structure of the Reduced n-Body Problem, nonlinearity, 2013, vol. 26, no. 6, pp. 1565–1579.MATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Fass`o, F. and Sansonetto, N., An Elemental Overview of the Nonholonomic Noether Theorem, Int. J. Geom. Methods Mod. Phys., 2009, vol. 6, no. 8, pp. 1343–1355.MathSciNetCrossRefGoogle Scholar
  41. 41.
    Garc´ia-Naranjo, L.C., Reduction of Almost Poisson Brackets for Nonholonomic Systems on Lie Groups, Regul. Chaotic Dyn., 2007, vol. 12, no. 4, pp. 365–388.MathSciNetCrossRefGoogle Scholar
  42. 42.
    Hamel, G., Die Lagrange-Eulerschen Gleichungen der Mechanik, Z. Math. u. Phys., 1904, vol. 50, pp. 1–57.MATHGoogle Scholar
  43. 43.
    Hamel, G., Über nichtholonome Systeme, Math. Ann., 1924, vol. 92, nos. 1–2, pp. 33–41.MATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Hermans, J., A Symmetric Sphere Rolling on a Surface, Nonlinearity, 1995, vol. 8, no. 4, pp. 493–515.MATHMathSciNetCrossRefGoogle Scholar
  45. 45.
    Hojman, S.A., The Construction of a Poisson Structure out of a Symmetry and a Conservation Law of a Dynamical System, J. Phys. A, 1996, vol. 29, no. 3, pp. 667–674.MATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    Jellett, J.H., A Treatise on the Theory on Friction, London: MacMillan, 1872.Google Scholar
  47. 47.
    Kim, B., Routh Symmetry in the Chaplygin’s Rolling Ball, Regul. Chaotic Dyn., 2011, vol. 16, no. 6, pp. 663–670.MATHMathSciNetCrossRefGoogle Scholar
  48. 48.
    Koiller, J., Reduction of Some Classical Non-Holonomic Systems with Symmetry, Arch. Rational Mech. Anal., 1992, vol. 118, no. 2, pp. 113–148.MATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    Koon, W. S. and Marsden, J.E., Poisson Reduction for Nonholonomic Mechanical Systems with Symmetry, Rep. Math. Phys., 1998, vol. 42, nos. 1–2, pp. 101–134.MATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    Kozlov, V.V., Dynamical Systems X: General Theory of Vortices, Encyclopaedia Math. Sci., vol. 67, Berlin: Springer, 2003.Google Scholar
  51. 51.
    Kozlov, V.V., On the Theory of Integration of the Equations of Nonholonomic Mechanics, Uspekhi Mekh., 1985, vol. 8, no. 3, pp. 85–107 (Russian).Google Scholar
  52. 52.
    Kozlov, V.V., Symmetries, Topology, and Resonances in Hamiltonian Mechanics, Ergeb. Math. Grenzgeb. (3), vol. 31, Berlin: Springer, 1996.Google Scholar
  53. 53.
    Kozlov, V.V., The Behaviour of Cyclic Variables in Integrable Systems, J. Appl. Math. Mech., 2013, vol. 77, no. 2, pp. 128–136; see also: Prikl. Mat. Mekh., 2013, vol. 77, no. 2, pp. 179–190.MATHMathSciNetCrossRefGoogle Scholar
  54. 54.
    Kozlov, V.V. and Kolesnikov, N.N., On Theorems of Dynamics, J. Appl. Math. Mech., 1978, vol. 42, no. 1, pp. 26–31; see also: Prikl. Mat. Mekh., 2013, vol. 77, no. 2, pp. 179–190.MathSciNetCrossRefGoogle Scholar
  55. 55.
    Lagrange, J. L., Mécanique analytique: Vol. 2, Cambridge: Cambridge Univ. Press, 2009.CrossRefGoogle Scholar
  56. 56.
    Laura, E., Sul moto parallelo ad un piano di un fluido in cui vi sono N vortici elementari, Atti della R. Acc. delle scienze di Torino, 1902, vol. 37, pp. 469–476.MATHGoogle Scholar
  57. 57.
    Lie, S., Begründung einer Invarianten-Theorie der Berührungs-Transformationen, Math. Ann., 1874, vol. 8, no. 2, pp. 215–303.MathSciNetCrossRefGoogle Scholar
  58. 58.
    Lie, S., Theorie der Transformationsgruppen: In 3 Vols., 2nd ed., Providence, R.I.: AMS, 1970.Google Scholar
  59. 59.
    Lie, S., Vorlesungen über Differentialgleichungen: Mit bekannten infinitesimalen Transformationen, Leipzig: Teubner, 1912.MATHGoogle Scholar
  60. 60.
    Neimark, Ju. I. and Fufaev, N.A., Dynamics of Nonholonomic Systems, Trans. Math. Monogr., vol. 33, Providence,R.I.: AMS, 1972.Google Scholar
  61. 61.
    Novikov, S.P. and Taimanov, I.A., Modern Geometric Structures and Fields, Grad. Stud. Math., vol. 71, Providence,R.I.: AMS, 2006.Google Scholar
  62. 62.
    Olver, P. J., Applications of Lie Groups to Differential Equations, 2nd ed., Grad. Texts in Math., vol. 107, Springer, 2000.Google Scholar
  63. 63.
    Ostrowski, J.P., The Mechanics and Control of Undulatory Robotic Locomotion, PhD Thesis, Pasadena: California Institute of Technology, 1996.Google Scholar
  64. 64.
    Poincaré, H., Les Méthodes Nouvelles de la Mécanique Céleste, Paris: Gauthier-Villars et fils, 1892.Google Scholar
  65. 65.
    Shchepetilov, A. V., Reduction of the Two-Body Problem with Central Interaction on Simply Connected Spaces of Constant Sectional Curvature, J. Phys. A, 1998, vol. 31, no. 29, pp. 6279–6291.MATHMathSciNetCrossRefGoogle Scholar
  66. 66.
    Smale, S., Topology and Mechanics: 1, Invent. Math., 1970, vol. 10, no. 4, pp. 305–331.MATHMathSciNetCrossRefGoogle Scholar
  67. 67.
    Souriau, J.-M., Structure des systèmes dynamiques, Paris: Dunod, 1970.MATHGoogle Scholar
  68. 68.
    Stückler, B., über die Berechnung der an rollenden Fahrzeugen wirkenden Haftreibungen, Arch. Appl. Mech., 1955, vol. 23, no. 4, pp. 279–287.Google Scholar
  69. 69.
    Stability of Motion: A Collection of Early Scientific Publications by E. J.Routh, W.K.Clifford, C. Sturm, and M. ocher, A.T. Fuller (Ed.), London: Taylor & Francis, 1975.Google Scholar
  70. 70.
    van der Schaft, A. J. and Maschke, B. M., On the Hamiltonian Formulation of Nonholonomic Mechanical Systems, Rep. Math. Phys., 1994, vol. 34, no. 2, pp. 225–233.MATHMathSciNetCrossRefGoogle Scholar
  71. 71.
    van Kampen, E.R. and Wintner, A., On a Symmetrical Canonical Reduction of the Problem of Three Bodies, Amer. J. Math., 1937, vol. 59, no. 1, pp. 153–166.MathSciNetCrossRefGoogle Scholar
  72. 72.
    Veselov, A.P. and Veselova, L. E., Integrable Nonholonomic Systems on Lie Groups, Math. Notes, 1988, vol. 44, no. 5, pp. 810–810; see also: Mat. Zametki, 1988, vol. 44, no. 5, pp. 604–618.MATHMathSciNetCrossRefGoogle Scholar
  73. 73.
    Volterra, V., Sopra una classe di equazioni dinamiche, Atti della R. Acc. delle scienze di Torino, 1897/98, vol. 33, pp. 451–475.Google Scholar
  74. 74.
    Woronetz, P.V., Transformation of the Equations of Motion by Means of Linear Integrals of Motion (with an Application to the n-Body Problem), Kiev. Univ. Izv., 1907. vol. 47, nos. 1–2, 192 pp. (Russian).Google Scholar
  75. 75.
    Whittaker, E.T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge: Cambridge Univ. Press, 1988.MATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Udmurt State UniversityIzhevskRussia
  2. 2.M. T. Kalashnikov Izhevsk State Technical UniversityIzhevskRussia

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