Regular and Chaotic Dynamics

, Volume 21, Issue 2, pp 232–248 | Cite as

Adiabatic invariants, diffusion and acceleration in rigid body dynamics

Article

Abstract

The onset of adiabatic chaos in rigid body dynamics is considered. A comparison of the analytically calculated diffusion coefficient describing probabilistic effects in the zone of chaos with a numerical experiment is made. An analysis of the splitting of asymptotic surfaces is performed and uncertainty curves are constructed in the Poincaré–Zhukovsky problem. The application of Hamiltonian methods to nonholonomic systems is discussed. New problem statements are given which are related to the destruction of an adiabatic invariant and to the acceleration of the system (Fermi’s acceleration).

Keywords

adiabatic invariants Liouville system transition through resonance adiabatic chaos 

MSC2010 numbers

70F15 37J30 37M25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Borisov, A. V. and Mamaev, I. S., Adiabatic Chaos in Rigid Body Dynamics, Regul. Chaotic Dyn., 1997, vol. 2, no. 2, pp. 65–78 (Russian).MathSciNetMATHGoogle Scholar
  2. 2.
    Neĭshtadt, A. I., Jumps in the Adiabatic Invariant on Crossing the Separatrix and the Origin of the 3: 1 Kirkwood Gap, Sov. Phys. Dokl., 1987, vol. 32, pp. 571–573; see also: Dokl. Akad. Nauk SSSR, 1987, vol. 295, pp. 47–50MATHGoogle Scholar
  3. 3.
    Arnol’d, V. I., Kozlov, V.V., and Neĭshtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, 3rd ed., Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 2006.Google Scholar
  4. 4.
    Wiggins, S., Adiabatic Chaos, Phys. Lett. A, 1988, vol. 128, nos. 6-7, pp. 339–342.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Neĭshtadt, A. I., Change of an Adiabatic Invariant at a Separatrix, Sov. J. Plasma Phys., 1986, vol. 12, pp. 568–573; see also: Fiz. Plazmy, 1986, vol. 12, no. 8, pp. 992–1000.Google Scholar
  6. 6.
    Tennyson, J. L., Cary, J.R., and Escande, D. F., Change of the Adiabatic Invariant due to Separatrix Crossing, Phys. Rev. Lett., 1986, vol. 56, no. 20, pp. 2117–2120.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Neĭshtadt, A. I., Sidorenko, V.V., and Treschev, D. V., Stable Periodic Motions in the Problem of Passage through a Separatix, Chaos, 1997, vol. 7, no. 1, pp. 2–11.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Kaper, T. J. and Kovačič, G., A Geometric Criterion for Adiabatic Chaos, J. Math. Phys., 1994, vol. 35, no. 3, pp. 1202–1218.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Rumer, Yu.B. and Ryvkin, M. Sh., Thermodynamics, Statistical Physics and Kinetics, Moscow: Mir, 1980.Google Scholar
  10. 10.
    Neĭshtadt, A. I., On the Change in the Adiabatic Invariant on Crossing a Separatrix in Systems with Two Degrees of Freedom, J. Appl. Math. Mech., 1987, vol. 51, no. 5, pp. 586–592; see also: Prikl. Mat. Mekh., 1987, vol. 51, no. 5, pp. 750–757.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Timofeev, A.V., On the Constancy of an Adiabatic Invariant when the Nature of the Motion Changes, JETP, 1978, vol. 48, no. 4, pp. 656–659; see also: Zh. Èksp. Teor. Fiz., 1978, vol. 75, no. 4, pp. 1303–1307.MathSciNetGoogle Scholar
  12. 12.
    Wisdom, J., A Perturbative Treatment of Motion near the 3/1 Commensurability, Icarus, 1985, vol. 63, no. 2, pp. 272–289.CrossRefGoogle Scholar
  13. 13.
    Neishtadt, A. N., Chaikovskii, D. K., and Chernikov, A. A., Adiabatic Chaos and Particle Diffusion, JETP, 1991, vol. 72, no. 3, pp. 423–430; see also: Zh. èksp. Teor. Fiz., 1991, vol. 99, no. 3, pp. 63–775.Google Scholar
  14. 14.
    Zhukovskii, N. E., Motion of a Rigid Body Containing a Cavity Filled with a Homogeneous Continuous Liquid, in Collected Works: Vol. 2, Moscow: Gostekhteorizdat, 1949, pp. 31–152 (Russian).Google Scholar
  15. 15.
    Borisov, A. V., On the Liouville Problem, in Numerical Modelling in the Problems of Mechanics, Moscow: Mosk. Gos. Univ., 1991, pp. 110–118 (Russian).Google Scholar
  16. 16.
    Neĭshtadt, A. I., Probability Phenomena due to Separatrix Crossing, Chaos, 1991, vol. 1, no. 1, pp. 42–48.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Aslanov, V. S., Integrable Cases in the Dynamics of Axial Gyrostats and Adiabatic Invariants, Nonlinear Dynam., 2012, vol. 68, nos. 1-2, pp. 259–273.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Elipe, A. and Lanchares, V., Exact Solution of a Triaxial Gyrostat with One Rotor, Celestial Mech. Dynam. Astronom., 2008, vol. 101, nos. 1-2, pp. 49–68.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Neĭshtadt, A. I., Passage through a Separatrix in a Resonance Problem with a Slowly-Varying Parameter, J. Appl. Math. Mech., 1975, vol. 39, no. 4, pp. 594–605; see also: Prikl. Mat. Mekh., 1975, vol. 39, no. 4, pp. 621–632.MathSciNetCrossRefGoogle Scholar
  20. 20.
    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.MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    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.MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    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.MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    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.MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    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.MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Hamiltonization of Nonholonomic Systems in the Neighborhood of Invariant Manifolds, Regul. Chaotic Dyn., 2011, vol. 16, no. 5, pp. 443–464.MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Gelfreich, V. and Turaev, D., Fermi Acceleration in Non-Autonomous Billiards, J. Phys. A, 2008, vol. 41, no. 21, 212003, 6 pp.MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Loskutov, A., Ryabov, A.B., and Akinshin, L. G., Properties of Some Chaotic Billiards with Time-Dependent Boundaries, J. Phys. A, 2000, vol. 33, no. 44, pp. 7973–7986.MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Kamphorst, S. O., Leonel, E. D., and da Silva, J. K. L., The Presence and Lack of Fermi Acceleration in Nonintegrable Billiards, J. Phys. A, 2007, vol. 40, no. 37, F887–F893.MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Lenz, F., Diakonos, F.K., and Schmelcher, P., Tunable Fermi Acceleration in the Driven Elliptical Billiard, Phys. Rev. Lett., 2008, vol. 100, no. 1, 014103, 4 pp.CrossRefGoogle Scholar
  30. 30.
    Koiller, J., Markarian, R., Kamphorst, S. O., and Pinto de Carvalho, S., Static and Time-Dependent Perturbations of the Classical Elliptical Billiard, J. Statist. Phys., 1996, vol. 83, nos. 1-2, pp. 127–143.MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Lichtenberg, A. J. and Lieberman, M.A., Regular and Chaotic Dynamics, 2nd ed., Appl. Math. Sci., vol. 38, New York: Springer, 1992.MATHCrossRefGoogle Scholar
  32. 32.
    Leonel, E.D. and Bunimovich, L.A., Suppressing Fermi Acceleration in a Driven Elliptical Billiard, Phys. Rev. Lett., 2010, vol. 104, no. 22, 224101, 4 pp.CrossRefGoogle Scholar
  33. 33.
    Bolotin, S. and Treschev, D., Unbounded Growth of Energy in Nonautonomous Hamiltonian Systems, Nonlinearity, 1999, vol. 12, no. 2, pp. 365–388.MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Gelfreich, V., Rom-Kedar, V., and Turaev, D., Fermi Acceleration and Adiabatic Invariants for Non- Autonomous Billiards, Chaos, 2012, vol. 22, no. 3, 033116, 21 pp.MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Shah, K., Gelfreich, V., Rom-Kedar, V., and Turaev, D., Leaky Fermi Accelerators, Phys. Rev. E, 2015, vol. 91, no. 6, 062920, 7 pp.CrossRefGoogle Scholar
  36. 36.
    Pereira, T. and Turaev, D., Fast Fermi Acceleration and Entropy Growth, Math. Model. Nat. Phenom., 2015, vol. 10, no. 3, pp. 31–47.MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Artemyev, A.V., Neishtadt, A. I., and Zelenyi, L. M., Rapid Geometrical Chaotization in Slow-Fast Hamiltonian Systems, Phys. Rev. E, 2014, vol. 89, no. 6, 060902, 4 pp.CrossRefGoogle Scholar
  38. 38.
    Leoncini, X., Kuznetsov, L., and Zaslavsky, G. M., Chaotic Advection Near a Three-Vortex Collapse, Phys. Rev. E, 2001, vol. 63, no. 3, 036224, 17 pp.CrossRefGoogle Scholar
  39. 39.
    Erdakova, N.N. and Mamaev, I. S., On the Dynamics of a Body with an Axisymmetric Base Sliding on a Rough Plane, Nelin. Dinam., 2013, vol. 9, no. 3, pp. 521–545 (Russian).CrossRefGoogle Scholar
  40. 40.
    Markeyev, A.P., The Equations of the Approximate Theory of the Motion of a Rigid Body with a Vibrating Suspension Point, J. Appl. Math. Mech., 2011, vol. 75, no. 2, 132–139; see also: Prikl. Mat. Mekh., 2011, vol. 75, no. 2, pp. 193–203.MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Yudovich, V. I., The Dynamics of a Particle on a Smooth Vibrating Surface, J. Appl. Math. Mech., 1998, vol. 62, no. 6, pp. 893–900; see also: Prikl. Mat. Mekh., 1998, vol. 62, no. 6, pp. 968–976.MathSciNetCrossRefGoogle Scholar
  42. 42.
    Markeyev, A.P., Approximate Equations of Rotational Motion of a Rigid Body Carrying a Movable Point Mass, J. Appl. Math. Mech., 2013, vol. 77, no. 2, pp. 137–144; see also: Prikl. Mat. Mekh., 2013, vol. 77, no. 2, pp. 191–201.MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Markeev, A.P., The Dynamics of a Rigid Body Colliding with a Rigid Surface, Regul. Chaotic Dyn., 2008, vol. 13, no. 2, pp. 96–129.MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Zaslavsky, G. M., Chaos in Dynamic Systems, New York: Harwood Academic Publishers, 1985.Google Scholar
  45. 45.
    Ivanova, T. B. and Pivovarova, E. N., Comments on the Paper by A.V.Borisov, A.A. Kilin, I. S.Mamaev “How To Control the Chaplygin Ball Using Rotors: 2”, Regul. Chaotic Dyn., 2014, vol. 19, no. 1, pp. 140–143.MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    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.MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Karavaev, Yu. L. and Kilin, A.A., The Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform, Regul. Chaotic Dyn., 2015, vol. 20, no. 2, pp. 134–152.MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Kilin, A.A., Pivovarova, E.N., and Ivanova, T.B., Spherical Robot of Combined Type: Dynamics and Control, Regul. Chaotic Dyn., 2015, vol. 20, no. 6, pp. 716–728.MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Borisov, A. V., Jalnine, A.Yu., Kuznetsov, S.P., Sataev, I.R., and Sedova, J. V., Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 512–532.MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Borisov, A. V., Kazakov, A.O., and Sataev, I. R., The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin’s Top, Regul. Chaotic Dyn., 2014, vol. 19, no. 6, pp. 718–733.MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Bizyaev, I. A., Borisov, A.V., and Kazakov, A.O., Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors, Regul. Chaotic Dyn., 2015, vol. 20, no. 5, pp. 605–626.MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    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., 2015, vol. 20, no. 5, pp. 605–626.MathSciNetCrossRefGoogle Scholar
  53. 53.
    Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Rolling of a Ball without Spinning on a Plane: The Absence of an Invariant Measure in a System with a Complete Set of Integrals, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 571–579.MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    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.MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    Kozlov, V.V. and Ramodanov, S. M., On the Motion of a Body with a Rigid Hull and Changing Geometry of Masses in an Ideal Fluid, Dokl. Phys., 2002, vol. 47, no. 2, pp. 132–135; see also: Dokl. Akad. Nauk, 2002, vol. 382, no. 4, pp. 478–481.MathSciNetCrossRefGoogle Scholar
  56. 56.
    Borisov, A. V. and Mamaev, I. S., On the Motion of a Heavy Rigid Body in an Ideal Fluid with Circulation, Chaos, 2006, vol. 16, no. 1, 013118, 7 pp.MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    Vetchanin, E. V. and Kilin, A.A., Free and Controlled Motion of a Body with Moving Internal Mass though a Fluid in the Presence of Circulation around the Body, Dokl. Phys., 2016, vol. 61, no. 1, pp. 32–36; see also: Dokl. Akad. Nauk, 2016, vol. 466, no. 3, pp. 293–297.CrossRefGoogle Scholar
  58. 58.
    Borisov, A. V. and Mamaev, I. S., The Rolling Motion of a Rigid Body on a Plane and a Sphere: Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 177–200.MathSciNetMATHCrossRefGoogle Scholar
  59. 59.
    Borisov, A. V., Mamaev, I. S., and Kilin, A.A., Rolling of a Ball on a Surface: New Integrals and Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 201–219.MathSciNetMATHCrossRefGoogle Scholar
  60. 60.
    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.MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    Borisov, A. V. andMamaev, I. S., Symmetries and Reduction in Nonholonomic Mechanics, Regul. Chaotic Dyn., 2015, vol. 20, no. 5, pp. 553–604.MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    Borisov, A. V. and Mamaev, I. S., Isomorphism and Hamilton Representation of Some Nonholonomic Systems, Siberian Math. J., 2007, vol. 48, no. 1, pp. 26–36; see also: Sibirsk. Mat. Zh., 2007, vol. 48, no. 1, pp. 33–45.MathSciNetMATHCrossRefGoogle Scholar
  63. 63.
    Borisov, A. V., Fedorov, Yu. N., and Mamaev, I. S., Chaplygin Ball over a Fixed Sphere: An Explicit Integration, Regul. Chaotic Dyn., 2008, vol. 13, no. 6, pp. 557–571.MathSciNetMATHCrossRefGoogle Scholar
  64. 64.
    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.MathSciNetMATHCrossRefGoogle Scholar
  65. 65.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support, Regul. Chaotic Dyn., 2012, vol. 17, no. 2, pp. 170–190.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

Personalised recommendations