Mathematical Notes

, Volume 95, Issue 3–4, pp 308–315 | Cite as

On the nonlinear Poisson bracket arising in nonholonomic mechanics

Article

Abstract

Nonholonomic systems describing the rolling of a rigid body on a plane and their relationship with various Poisson structures are considered. The notion of generalized conformally Hamiltonian representation of dynamical systems is introduced. In contrast to linear Poisson structures defined by Lie algebras and used in rigid-body dynamics, the Poisson structures of nonholonomic systems turn out to be nonlinear. They are also degenerate and the Casimir functions for them can be expressed in terms of complicated transcendental functions or not appear at all.

Keywords

Poisson bracket nonholonomic system Poisson structure dynamical system conformally Hamiltonian representation Casimir function Routh sphere rolling of a Chaplygin ball 

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Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  • A. V. Borisov
    • 1
    • 2
    • 3
  • I. S. Mamaev
    • 1
    • 2
    • 4
  • A. V. Tsyganov
    • 5
  1. 1.Udmurt State UniversityIzhevskRussia
  2. 2.Machine Construction InstituteRussian Academy of SciencesMoscowRussia
  3. 3.Institute of Mathematics and Mechanics, Ural BranchRussian Academy of SciencesEkaterinburgRussia
  4. 4.Russian Academy of SciencesEkaterinburgRussia
  5. 5.St. Petersburg State UniversitySt. PetersburgRussia

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