Isomorphism and Hamilton representation of some nonholonomic systems
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We consider some questions connected with the Hamiltonian form of the two problems of nonholonomic mechanics: the Chaplygin ball problem and the Veselova problem. For these problems we find representations in the form of the generalized Chaplygin systems that can be integrated by the reducing multiplier method. We give a concrete algebraic form of the Poisson brackets which, together with an appropriate change of time, enable us to write down the equations of motion of the problems under study. Some generalization of these problems are considered and new ways of implementation of nonholonomic constraints are proposed. We list a series of nonholonomic systems possessing an invariant measure and sufficiently many first integrals for which the question about the Hamiltonian form remains open even after change of time. We prove a theorem on isomorphism of the dynamics of the Chaplygin ball and the motion of a body in a fluid in the Clebsch case.
Keywordsnonholonomic system reducing multiplier Hamiltonization isomorphism
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- 1.Chaplygin S. A., “To the theory of motion of nonholonomic systems. The theorem of a reducing multiplier,” in: Collected Works. Vol. 1 [in Russian], Gostekhizdat, Moscow; Leningrad, 1948, pp. 15–25.Google Scholar
- 2.Fedorov Yu. N. and Jovanović B., “Nonholonomic LR systems as generalized Chaplygin systems with an invariant measure and flows on homogeneous spaces,” J. Nonlinear Sci., 4, No. 14, 341–381 (2004).Google Scholar
- 3.Chaplygin S. A., “On the rolling of a ball on a horizontal plane,” in: Collected Works. Vol. 1 [in Russian], Gostekhizdat, Moscow; Leningrad, 1948, pp. 78–101.Google Scholar
- 5.Markeev A. P., “Integrability of the ball rolling problem with a multiply connected cavity filled with an ideal fluid,” Izv. Akad. Nauk SSSR Mekh. Tverd. Tela, 21, No. 1, 64–65 (1986).Google Scholar
- 6.Arnol’d V. I., Kozlov V. V., and Neishtadt A. I., Mathematical Aspects of Classical and Celestial Mechanics [in Russian], Éditorial, URSS, Moscow (2002).Google Scholar
- 7.Suslov G. K., Theoretical Mechanics [in Russian], GTTI, Moscow; Leningrad (1948).Google Scholar
- 8.Veselova L. E., “New cases of integrability of the equations of motion of a rigid body in the presence of a nonholonomic constraint,” in: Geometry, Differential Equations, and Mechanics [in Russian], Moscow Univ., Moscow, 1986, pp. 64–68.Google Scholar
- 12.Fedorov Yu. N., “On two integrable nonholonomic systems in the classical dynamics,” Vestnik MGU Ser. Mat. Mekh., 4, 38–41 (1989).Google Scholar
- 14.Kozlov V. V., Symmetries, Topology, and Resonances in Hamiltonian Mechanics [in Russian], Izdat. UdGU, Izhevsk (1995).Google Scholar
- 17.Duistermaat J. J., “Chaplygin’s sphere,” in: Cushman R., Duistermaat J. J., and Śniatycki J. Chaplygin and the Geometry of Nonholonomically Constrained Systems (2000). E-print arXiv:[math/040919]arxiv.org/abs/math.psGoogle Scholar
- 18.Ehlers K., Koiller J., Montgomery R., and Rios P., “Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization,” in: The Breadth of Symplectic and Poisson Geometry (Eds. J. E. Marsden, T. S. Ratiu), Festschrift in Honor of Alain Weinstein, Birkhäuser, Boston, 2005 (Progr. Math.; V. 232).Google Scholar