Regular and Chaotic Dynamics

, Volume 13, Issue 5, pp 443–490 | Cite as

Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems

Nonholonomic Mechanics


This paper can be regarded as a continuation of our previous work [1, 2] on the hierarchy of the dynamical behavior of nonholonomic systems. We consider different mechanical systems with nonholonomic constraints; in particular, we examine the existence of tensor invariants (laws of conservation) and their connection with the behavior of a system. Considerable attention is given to the possibility of conformally Hamiltonian representation of the equations of motion, which is mainly used for the integration of the considered systems.

Key words

nonholonomic systems implementation of constraints conservation laws hierarchy of dynamics explicit integration 

MSC2000 numbers

34D20 70E40 37J35 


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  1. 1.
    Borisov, A.V. and Mamaev, I.S., The Rolling of Rigid Body on a Plane and Sphere. Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 1, pp. 177–200.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    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. 201–220.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Ferrers, N.M., Extension of Lagrange’s Equations, Quart. J. Pure Appl. Math., 1872, vol. 12, no. 45, pp. 1–5.Google Scholar
  4. 4.
    Sumbatov, A.S., Nonholonomic Systems, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 221–238.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Benenti, S., A “User-Friendly” Approach to the Dynamical Equations of Nonholonomic Systems, SIGMA, 2007, vol. 3, 036 (33 pages).Google Scholar
  6. 6.
    Woronetz, P.V., On Equations of Motion of Nonholonomic Systems, Matematicheskii sbornik (Mathematical Collection), 1901, vol. 22, no. 4, pp. 659–686 (in Russian).Google Scholar
  7. 7.
    Woronetz, P.V., Equations of Motion of a Rigid Body Rolling along a Stationary Surface Without Slipping, Kiev: Proc. of Kiev University, 1903, vol. 43, no. 1, pp. 1–66.Google Scholar
  8. 8.
    Woronetz, P.V., Transformation of Equations of Motion with the Help of Linear Integrals (with Application to the 3-Body Problem), Kiev: Proc. of Kiev University, 1907, vol. 47, no. 1–2, pp. 1–192.Google Scholar
  9. 9.
    Woronetz, P., Über die rollende Bewegung einer Kreisscheibe auf einer beliebigen Flache unter der Wirkung von gegebenen Kraften Math. Annalen, 1909, vol. 67, pp. 268–280.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Woronetz, P., Über die Bewegung eines starren Körpers, der ohne Gleitung auf einer beliebigen Fläche rollt, Math. Annalen, 1911, vol. 70, pp. 410–453.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Woronetz, P., Über die Bewegungsgleichungen eines starren Körpers, Math. Annalen, 1912, vol. 71, pp. 392–403.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Arnold V.I., Kozlov V.V., and Neishtadt A.I. Mathematical Aspects of Classical and Celestial Mechanics, Itogi Nauki i Tekhniki. Sovr. Probl. Mat. Fundamental’nye Napravleniya, Vol. 3, VINITI, Moscow 1985. English transl.: Encyclopadia of Math. Sciences, Vol. 3, Berlin: Springer-Verlag, 1989.Google Scholar
  13. 13.
    Borisov, A.V. and Mamaev, I.S., Chaplygin’s Ball. The Suslov Problem and Veselova’s Problem. Integrability and Realization of Constraints, in (Nonholonomic Dynamical Systems), Borisov, A.V. and Mamaev, I.S., Eds., Moscow-Izhevsk: RCD, Institute of Computer Sciences, 2002, pp. 118–130.Google Scholar
  14. 14.
    Ehlers, K. and Koiller, J., Rubber Rolling: Geometry and Dynamics of 2-3-5 Distributions, in Proceedings IUTAM symposium 2006 on Hamiltonian Dynamics, Vortex Structures, Turbulence (Moscow, 25–30 August, 2006), pp. 469–480.Google Scholar
  15. 15.
    Chaplygin, S.A., On a Ball’s Rolling on a Horizontal Plane, Matematicheskiĭ sbornik (Mathematical Collection), 1903, vol. 24. [English translation: Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 131–148.]
  16. 16.
    Veselova, L.E., New Cases of the Integrability of the Equations of Motion of a Rigid Body in the Presence of a Nonholonomic Constraint, in Geometry, Differential Equations and Mechanics, Moskov. Gos. Univ., Mekh.-Mat. Fak., Moscow, 1986, pp. 64–68.Google Scholar
  17. 17.
    Rashevsky, P.K., About Connecting Two Points of a Completely Nonholonomic Space by Admissible Curve, Uch. Zap. Ped. Inst. im. Liebknechta, Ser. Phys. Math., 1938, no. 2, pp. 83–94.Google Scholar
  18. 18.
    Chow, W.L., Über Systeme von linearen partiellen Differential Gleichungen erster Ordnung, Math. Ann., 1939, vol. 117, pp. 98–105.CrossRefMathSciNetGoogle Scholar
  19. 19.
    Agrachev, A.A., Rolling Balls and Octonions, Proc. Steklov Institute of Mathematics, 2007, vol. 258, pp. 13–22 [Tr. Mat. Inst. Steklova, 2007, vol. 258, pp. 17–27].CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Bor, G. and Montgomery, R., G 2 and the “Rolling Distribution”, arXiv:math.DG/0612469v1.Google Scholar
  21. 21.
    Chaplygin, S.A., On the Theory of Motion of Nonholonomic Systems. Example of Application of the Reducing Mutiplier Method, unpublished note; printed in Collected Papers, Moscow-Leningrad: Gostekhizdat, 1948, Vol. 3, pp. 248–275.Google Scholar
  22. 22.
    Suslov, G.K., Teoreticheskaya mekhanika (Theoretical Mechanics), Moscow-Leningrad: Gostekhizdat, 1946.Google Scholar
  23. 23.
    Zhuravlev, V.F., On a Model of a Dry Friction in Problems of Rigid Body Dynamics, Adv. in Mech., 2005, no. 3, pp. 58–76 (in Russian).Google Scholar
  24. 24.
    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 [Prikl. Mat. Mekh., 1987, vol. 51, no. 4, pp. 538–545].MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Kolmogorov, A.N., On Dynamical Systems with an Integral Invariant on the Torus, Doklady Akad. Nauk SSSR, 1953, vol. 93, pp. 763–766 (in Russian).MATHMathSciNetGoogle Scholar
  26. 26.
    Borisov, A.V. and Mamaev, I.S., Strange Attractors in Rattleback Dynamics, Uspehi Fiz. Nauk, 2003, vol. 173, no. 4, pp. 407–418 [Physics-Uspekhi, 2003, vol. 46, no. 4, pp. 393–403].MathSciNetGoogle Scholar
  27. 27.
    Veselov, A.P. and Veselova, L.E., Integrable Nonholonomic Systems on Lie Groups, Math. Notes, 1988, vol. 44, no. 5–6, pp. 810–819 [Mat. Zametki, 1988, vol. 44, no. 5, pp. 604–619].MATHMathSciNetGoogle Scholar
  28. 28.
    Kozlov, V.V., On the Integration Theory of Equations of Nonholonomic Mechanics, Adv. in Mech., 1985, vol. 8, no. 3, pp. 85–107 (in Russian). See also: Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 191–176.MathSciNetGoogle Scholar
  29. 29.
    Borisov, A.V. and Mamaev, I.S., Puassonovy struktury i algebry Li v gamil’tonovoi mekhanike (Poisson Structures and Lie Algebras in Hamiltonian Mechanics), vol. 7 of Library “R & C Dynamics”, Izhevsk, 1999.Google Scholar
  30. 30.
    Tatarinov, Ya.V., Separation of Variables and New Topological Phenomena in Holonomic and Nonholonomic Systems, Trudy Sem. Vektor. Tenzor. Anal., 1988, no. 23, pp. 160–174 (in Russian).Google Scholar
  31. 31.
    Novikov, S.P. and Taimanov, I.A., Modern Geometric Structures and Fields, vol. 71 of Graduate Studies in Mathematics, Providence, RI: AMS, 2006.MATHGoogle Scholar
  32. 32.
    Olver, P.J., Applications of Lie Groups to Differential Equations, 2nd ed., vol. 107 of Graduate Texts in Mathematics, New York: Springer-Verlag, 1993.MATHGoogle Scholar
  33. 33.
    Painlevé, P., Lecons sur le frottement, P.: Hermann, 1895.MATHGoogle Scholar
  34. 34.
    Weinstein, A., Poisson Geometry, Diff. Geom. Appl., 1998, vol. 9, no. 1–2, pp. 213–238.CrossRefGoogle Scholar
  35. 35.
    Borisov, A.V. and Dudoladov, S.L., Kovalevskaya Exponents and Poisson Structures, Regul. Chaotic Dyn., 1999, vol. 4, no. 3, pp. 13–20.MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Kozlov, V.V., Symmetries, Topology and Resonances in Hamiltonian Mechanics, Berlin: Springer-Verlag, 1996.Google Scholar
  37. 37.
    Borisov, A.V. and Mamaev, I.S., Dinamika tverdogo tela. Gamiltonovy metody, integriruemost’, khaos (Rigid Body Dynamics. Hamiltonian Methods, Integrability, Chaos), Moscow-Izhevsk: Inst. komp. issled., RCD, 2005.Google Scholar
  38. 38.
    Chaplygin, S.A., On a Motion of a Heavy Body of Revolution on a Horizontal Plane, Trudy Otdeleniya fizicheskikh nauk Obshchestva lyubiteleĭ estestvoznaniya (Transactions of the Physical Section of Moscow Society of Friends of Natural Scientists), 1897, vol. 9, no. 1 [English translation: Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 119–130;]
  39. 39.
    Gallop, M.A., On the Rise of a Spinning Top, Trans. Cambridge Phil. Society, 1904, vol. 19, pp. 356–373.Google Scholar
  40. 40.
    Ramos, A., Poisson Structures for Reduced Non-holonomic Systems, J. Phys. A, 2004, vol. 37, no. 17, pp. 4821–4842.MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Fassò, F., Giacobbe, A., and Sansonetto, N., Periodic Flows, Rank-Two Poisson Structures, and Nonholonomic Mechanics, Regul. Chaotic Dyn., 2005, vol. 10, no. 3, pp. 267–284.MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Tatarinov, Ya.V., Construction of Compact Invariant Manifolds, not Diffeomorphic to Tori, in One Integrable Nonholonomic Problem, Uspekhi Matem. Nauk, no. 5, page 216 (in Russian).Google Scholar
  43. 43.
    Bates, L. and Cushman, R., What is a Completely Integrable Nonholonomic Dynamical System? Rep. on Math. Phys., 1999, vol. 44, no. 1–2, pp. 29–35.MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Sinai, Ya.G., Introduction to Ergodic Theory, vol. 18 of Mathematical Notes, Princeton, N.J.: Princeton University Press, 1976.MATHGoogle Scholar
  45. 45.
    Kornfel’d, I.P., Sinai, Ya.G., and Fomin, S.V., Ergodicheskaya teoriya (Ergodic Theory), Moscow: Nauka, 1980.Google Scholar
  46. 46.
    Borisov, A.V. and Mamaev, I.S., Obstacle to the Reduction of Nonholonomic Systems to the Hamiltonian Form, Dokl. Phys., 2002, vol. 47, no. 12, pp. 892–894 [Dokl. Akad. Nauk, 2002, vol. 387, no. 6, pp. 764–766].CrossRefMathSciNetGoogle Scholar
  47. 47.
    Kozlov, V.V., Diffusion in Systems with Integral Invariants on the Torus, Dokl. Math., 2001, vol. 64, no. 3, pp. 390–392 [Dokl. Akad. Nauk, 2001, vol. 381, no. 5, pp. 596–598].Google Scholar
  48. 48.
    Bolsinov, A.V. and Fomenko, A.T., Integrable Hamiltonian Systems: Geometry, Topology, Classification, CRC Press, 2004.Google Scholar
  49. 49.
    Chaplygin, S.A., On the Theory of Motion of Nonholonomic Systems. The Reducing-Multiplier Theorem, Matematicheskii sbornik (Mathematical Collection), 1911, vol. 28, no. 1 [English translation: Regul. Chaotic Dyn., 2008, vol. 13, no. 4, pp. 369–376;].
  50. 50.
    Cantrijin, F., de Léon, M., and de Diego, D., On the Geometry of Generalized Chaplygin Systems, Math. Proc. Camb. Phil. Soc., 2002, vol. 132, pp. 323–351.Google Scholar
  51. 51.
    Koiller, J., Reduction of Some Classical Nonholonomic Systems with Symmetry, Arch. Rational. Mech. Anal., 1992, vol. 118, pp. 113–148.MATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Fedorov, Yu.N. and Jovanović, B., Nongholonomic LR-Systems as Generalized Chaplygin Systems with an Invariant Measure and Flows on Homogeneous Spaces, J. of Nonlinear Science, 2004, vol. 14, pp. 341–381.MATHGoogle Scholar
  53. 53.
    Moshchuk, N.K., Reducing the Equations of Motion of Certain Nonholonomic Chaplygin Systems to Lagrangian and Hamiltonian Form, J. Appl. Math. Mech., 1987, vol. 51, no. 2, pp. 172–177 [Prikl. Mat. Mekh., 1987, vol. 51, no. 2, pp. 223–229].MATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Rumyantsev, V.V. and Sumbatov, A.V., On the Problem of Generalization of the Hamilton-Jacobi Method for Non-holonomic System, ZAMM, 1978, vol. 58, pp. 477–781.MATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    Dragovic, V., Gajic, B., and Jovanovic, B., Generalizations of Classical Integrable Nonholonomic Rigid Body Systems, J. Phys. A: Math. Gen., 1998, vol. 31, pp. 9861–9869.MATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    Kharlamova-Zabelina, E.I., Rapid Rotation of a Rigid Body about a Fixed Point under the Presence of a Nonholonomic Constraint, Vestnik Moskov. Univ. Ser. Mat. Mekh. Astron. Fiz. Khim., 1957, vol. 12, no. 6, pp. 25–34 (in Russian).Google Scholar
  57. 57.
    Kharlamov, A.P., The Inertial Motion of a Body with a Fixed Point and Subject to a Nonholonomic Constraint, Mekh. Tverd. Tela, Donetsk, 1995, no. 27, pp. 21–31 (in Russian).Google Scholar
  58. 58.
    Bloch, A.M., Nonholonomic Mechanics and Control. With the collaboration of Baillieul, J., Crouch, P., and Marsden, J. With scientific input from Krishnaprasad, P.S., Murray, R.M., and Zenkov, D., vol. 24 of Interdisciplinary Applied Mathematics, Systems and Control, New York: Springer-Verlag, 2003.Google Scholar
  59. 59.
    Fedorov, Yu.N., Two Integrable Nonholonomic Systems in Classical Dynamics, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1989, no. 4, pp. 38–41 (in Russian).Google Scholar
  60. 60.
    Kilin, A.A., The Dynamics of Chaplygin ball: the Qualitative and Computer Analisis, Regul. Chaotic Dyn., 2001, vol. 6, no. 3, pp. 291–306.MATHCrossRefMathSciNetGoogle Scholar
  61. 61.
    Schneider, D.A., Non-holonomic Euler-Poincaré Equations and Stability in Chaplygin’s Sphere, Dyn. Sys., 2002, vol. 17, no. 2, pp. 87–130.MATHCrossRefGoogle Scholar
  62. 62.
    Markeev, A.P., Integrability of a Problem on Rolling of Ball with Multiply Connected Cavity Filled by Ideal Liquid, Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, 1986, vol. 21, no. 1, pp. 64–65 (in Russian).Google Scholar
  63. 63.
    Duistermaat, J.J., Chaplygin’s Sphere, in Cushman, R., Duistermaat, J.J., and Śniatycki, J., Chaplygin and the Geometry of Nonholonomically Constrained Systems (in preparation), 2000; arXiv:math.DS/0409019.Google Scholar
  64. 64.
    Borisov, A.V. and Mamaev, I.S., The Chaplygin Problem of the Rolling Motion of a Ball Is Hamiltonian, Math. Notes, 2001, vol. 70, no. 5, pp. 720–723 [Mat. Zametki, 2001, vol. 70, no. 5, pp. 793–796].MATHCrossRefMathSciNetGoogle Scholar
  65. 65.
    Ehlers, K., Koiller, J., Montgomery, R., and Rios, P.M., Nonholonomic Systems via Moving Frames: Cartan Equivalence and Chaplygin Hamiltonization. The Breadth of Symplectic and Poisson Geometry, vol. 232 of Progr. Math., Boston, MA: Birkhäuser Boston, 2005, pp. 75–120.Google Scholar
  66. 66.
    Garcia-Naranjo, L., Reduction of Almost Poisson Brackets for Nonholonomic Systems on Lie Groups, Regul. Chaotic Dyn., 2007, vol. 12, no. 4, pp. 365–388.CrossRefMathSciNetGoogle Scholar
  67. 67.
    Kozlov, V.V. and Fedorov, Yu.N., A Memoir on Integrable Systems, Springer-Verlag (in preparation).Google Scholar
  68. 68.
    Borisov, A.V. and Fedorov, Y.N., On Two Modified Integrable Problems of Dynamics, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1995, no. 6, pp. 102–105 (in Russian).Google Scholar
  69. 69.
    Contensou, P., Couplage entre frottement de glissement et frottement de pivotement dans la théorie de la toupie, Kreiselprobleme Gydrodynamics: IUTAM Symp. Celerina, Berlin: Springer, 1963, pp. 201–216.Google Scholar
  70. 70.
    Yaroshchuk, V.A., New Cases of the Existence of an Integral Invariant in a Problem on the Rolling of a Rigid Body, Without Slippage, on a Fixed Surface, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1992, no. 6, pp. 26–30 (in Russian).Google Scholar
  71. 71.
    Borisov, A.V. and Mamaev, I.S., Rolling of a Non-homogeneous Ball over a Sphere Without Slipping and Twisting, Regul. Chaotic Dyn., 2007, vol. 12, no. 2, pp. 153–159.CrossRefMathSciNetGoogle Scholar
  72. 72.
    Borisov, A.V., Mamaev I.S., and Marikhin, V.G., Explicit Integration of One Problem in Nonholonomic Mechanics, Dokl. Akad. Nauk, 2008 (in press).Google Scholar
  73. 73.
    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 [Sibirsk. Mat. Zh., 2007, vol. 48, no. 1, pp. 33–45]; Scholar
  74. 74.
    Marikhin, V.G. and Sokolov, V.V., Pairs of Commuting Hamiltonians that are Quadratic in Momenta, Theoret. and Math. Phys., 2006, vol. 149, no. 2, pp. 1425–1436 [Teoret. Mat. Fiz., 2006, vol. 149, no. 2, pp. 147–160].CrossRefMathSciNetMATHGoogle Scholar
  75. 75.
    Eisenhart, L.P., Separable Systems of Stäckel, Annals of Mathematics, 1934, vol. 35, no. 2, pp. 284–305.CrossRefMathSciNetGoogle Scholar
  76. 76.
    Ehlers, K. and Koiller, J., Rubber Rolling over a Sphere, Regul. Chaotic Dyn., 2007, vol. 12, no. 2, pp. 127–152.CrossRefMathSciNetGoogle Scholar
  77. 77.
    Braden, H.W., A Completely Integrable Mechanical System, Lett. in Math. Phys., 1982, vol. 6, pp. 449–452.MATHCrossRefMathSciNetGoogle Scholar
  78. 78.
    Poincaré, H., On Curves Defined by Differential Equations, Moscow-Leningrad: Gostekhizdat, 1947.Google Scholar
  79. 79.
    Routh, E., Dynamics of a System of Rigid Bodies. Part II, New York: Dover Publications, 1905.Google Scholar
  80. 80.
    Markeev, A.P., On the Dynamics of a Top, Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, 1986, no. 3, pp. 30–38 (in Russian).Google Scholar
  81. 81.
    Karapetyan, A.V., On the Realization of Non-holonomic Constraints by Forces of Viscous Friction and the Stability of Celtic Stones, J. Appl. Math. Mech., 1982, vol. 45, pp. 30–36 [Prikl. Mat. Mekh., 1981, vol. 45, pp. 42–51].CrossRefGoogle Scholar
  82. 82.
    Moshchuk, N.K., On the Motion of the Chaplygin Ball on a Hotisontal Plane, Prikl. Mat. Mekh., 1983, vol. 47, no. 6, pp. 916–921.Google Scholar
  83. 83.
    Moshchuk, N.K., Qualitative Analysis of the Motion of a Rigid Body of Revolution on an Absolutely Rough Plane, J. Appl. Math. Mech., 1988, vol. 52, no. 2, pp. 159–165 [Prikl. Mat. Mekh., 1988, vol. 52, no. 2, pp. 203–210].MATHCrossRefMathSciNetGoogle Scholar
  84. 84.
    Jensen, E.T. and Shegelski, M.R.A., The Motion of Curling Rocks: Experimental Investigation and Semi-Phenomenological Description, Can. J. Phys./Rev. Can. Phys., 2004, vol. 82, no. 10, pp. 791–809.CrossRefGoogle Scholar
  85. 85.
    Persson, B.N.J., Sliding Friction — Physical Principals and Applications, 2nd ed., London: Springer, 2000.Google Scholar
  86. 86.
    Neimark, Yu.I. and Fufaev, N.A., Dynamics of Nonholonomic Systems, vol. 33 of Trans. of Math. Mon., 1972.Google Scholar
  87. 87.
    Erismann, Th., Theorie und Anwendungen des echten Kugelgetriebes, Z. angew. Math. Phys., 1954, vol. 5, pp. 355–388.MATHCrossRefMathSciNetGoogle Scholar
  88. 88.
    Zhuravlev, V.F., On a Model of Dry Friction in the Problem of the Rolling of Rigid Bodies, J. Appl. Math. Mech., 1998, vol. 62, no. 5, pp. 705–710 [Prikl. Mat. Mekh., 1998, vol. 62, no. 5, pp. 762–767].CrossRefMathSciNetGoogle Scholar
  89. 89.
    Zhuravlev, V.F., Dynamics of a Heavy Homogeneous Body on a Rough Plane, Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, 2006, no. 6, pp. 3–8 (in Russian).Google Scholar
  90. 90.
    Zhuravlev, V.F. and Klimov, D.M., Global Motion of a Rattleback, Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, 2008, no. 3, pp. 8–16 (in Russian).Google Scholar
  91. 91.
    Leine, R.I., Le Saux, C., and Glocker, C., Friction Models for the Rolling Disk, ENOC-2005, Eindhoven, Netherlands, August, 2005.Google Scholar
  92. 92.
    Kozlov, V.V., Realization of Nonintegrable Constraints in Classical Mechanics, Soviet Phys. Dokl., 1983, vol. 28, no. 9, pp. 735–737 [Dokl. Akad. Nauk SSSR, 1983, vol. 272, no. 3, pp. 550–554].MATHGoogle Scholar
  93. 93.
    Argatov, I.I., Equilibrium conditions for a rigid body on a rough plane in the case of axially symmetric distribution of normal pressures, Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, 2005, no. 2, pp. 16–26 (in Russian).Google Scholar
  94. 94.
    Farkas, Z., Bartels, G., Unger, T., and Wolf, D.E., Frictional Coupling between Sliding and Spinning Motion, Phys. Rev. Let., 2003, vol. 90, no. 24, 248302 (4 pages).Google Scholar
  95. 95.
    Zhuravlev, V.F. and Klimov, D.M., On the Dynamics of the Thompson Top (Tippe Top) on a Plane with Real Dry Friction, Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, 2005, no. 6, pp. 157–168 (in Russian).Google Scholar
  96. 96.
    Pfeiffer, F., Einführung in die Dynamik (Introduction to dynamics), Teubner Studienbücher Mechanik [Teubner Mechanics Textbooks], Stuttgart: B.G. Teubner, 1989.MATHGoogle Scholar
  97. 97.
    Jellet, J.H., A Treatise on the Theory of Friction, London: Macmillan, 1872.Google Scholar
  98. 98.
    Markeev, A.P., Dinamika tela, soprikasayushchegosya s tverdoi poverkhnost’yu (The Dynamics of a Body Contiguous to a Solid Surface), Moscow: Nauka, 1991.Google Scholar
  99. 99.
    Kozlov, V.V. and Kolesnikov, N.N., On Theorems of Dynamics, J. Appl. Math. Mech., 1978, vol. 42, no. 1, pp. 28–33 (in Russian).CrossRefMathSciNetGoogle Scholar
  100. 100.
    Borisov, A.V. and Mamaev, I.S., An Integrable System with a Nonintegrable Constraint, Math. Notes, 2006, vol. 80, no. 1–2, pp. 127–130 [Mat. Zametki, 2006, vol. 80, no. 1, pp. 131–134].MATHCrossRefMathSciNetGoogle Scholar
  101. 101.
    Kessler, P. and O’Reilly, O.M., The Ringing of Euler’s Disk, Regul. Chaotic Dyn., 2002, vol. 7, pp. 49–60.MATHCrossRefMathSciNetGoogle Scholar
  102. 102.
    Appell, P., Traité de mécanique rationelle, Paris: Gauthier-Villars, 1896.Google Scholar
  103. 103.
    Coriolis, G., Théorie mathématique des effects du jeu de Billard, Paris: Carilian-Goeury, 1835.Google Scholar
  104. 104.
    Ivanov, A.P., On Detachment Conditions in the Problem on the Motion of a Rigid Body on a Rough Plane, Regul. Chaotic Dyn., 2008, vol. 13, no. 4, pp. 355–368.CrossRefMathSciNetGoogle Scholar
  105. 105.
    Ivanov, A.P., Geometric Representation of Detachment Conditions in a System with Unilateral Constraint, Regul. Chaotic Dyn., 2008, vol. 13, no. 5, pp. 436–443.CrossRefGoogle Scholar
  106. 106.
    Fedorov, Yu.N., New Cases of the Integrability of the Equations of Motion of a Rigid Body in the Presence of a Nonholonomic Constraint, in Geometry, Differential Equations and Mechanics, Moscow: Moskov. Gos. Univ., Mekh.-Mat. Fak., 1986, pp. 151–155.Google Scholar
  107. 107.
    Kotter, F., Ueber die Bewegung eines festen Korpers in einer Flussigkeit, Z. angew. Math. Phys., 1892, vol. 109, Part I: pp. 51–81, Part II: pp. 89–111.Google Scholar
  108. 108.
    Sokolov, V.V. and Marikhin, V.G., Separation of Variables on a Non-hyperelliptic Curve, Regul. Chaotic Dyn., 2005, vol. 10, pp. 59–70.MATHCrossRefMathSciNetGoogle Scholar
  109. 109.
    Marikhin, V.G. and Sokolov, V.V., On the Reduction of the Pair of Hamiltonians Quadratic in Momenta to Canonic Form and Real Partial Separation of Variables for the Clebsch Top, Rus. J. Nonlin. Dyn., 2008, vol. 4, no. 3 (in press).Google Scholar
  110. 110.
    Fedorov, Yu.N., The Motion of a Rigid Body in a Spherical Support, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1988, no. 5, pp. 91–93 (in Russian).Google Scholar
  111. 111.
    Kharlamov, A.P., Topologicheskii analiz integriruemykh zadach dinamiki tverdogo tela (Topological Analysis of Integrable Problems of Rigid Body Dynamics), Leningrad. Univ., 1988.Google Scholar
  112. 112.
    Fedorov, Yu.N., Dynamic Systems with the Invariant Measure on Riemann’s Symmetric Pairs (gl(n), so(n)), Regul. Chaotic Dyn., 1996, vol. 1, no. 1, pp. 38–44.MATHGoogle Scholar
  113. 113.
    Fedorov, Yu.N. and Jovanović, B., Quasi-Chaplygin Systems and Nonholonomic Rigid Body Dynamics, Lett. Math. Phys., 2006, vol. 76, pp. 215–230.MATHCrossRefMathSciNetGoogle Scholar
  114. 114.
    Fedorov, Yu.N. and Kozlov, V.V., Various Aspects of n-Dimensional Rigid Body Dynamics, vol. 168 of Amer. Math. Soc. Transl. (2), 1995, pp. 141–171.Google Scholar
  115. 115.
    Moiseev, N.N. and Rumyantsev, V.V., Dinamika tela s polostyami, soderzhaschimi zhidkost’ (Dynamics of a Body with Cavities Filled with Liquid), Moscow: Nauka, 1965.Google Scholar
  116. 116.
    Karapetyan, A.V. and Prokonina, O.V., On the Stability of Uniform Rotations of a Top with a Fluid-Filled Cavity on a Plane with Friction, J. Appl. Math. Mech., 2000, vol. 64, no. 1, pp. 81–86 [Prikl. Mat. Mekh., 2000, vol. 64, no. 1, pp. 85–91].CrossRefMathSciNetGoogle Scholar

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© MAIK Nauka 2008

Authors and Affiliations

  1. 1.Institute of Computer ScienceUdmurt State UniversityIzhevskRussia

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