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.
Poisson bracket, nonholonomic system, Poisson structure, dynamical system, con- formally Hamiltonian representation, Casimir function, Routh sphere, rolling of a Chaplygin ball
Borisov A. V., Mamaev I. S., Tsiganov A. V., On the Nonlinear Poisson Bracket Arising in Nonholonomic Mechanics , Mathematical Notes, 2014, vol. 95, no. 3, pp. 308-315
This is a survey of basic facts presently known about non-linear Poisson structures in the analysis of integrable systems in non-holonomic mechanics. It is shown that by using the theory of Poisson deformations it is possible to reduce various non-holonomic systems to dynamical systems on well-understood phase spaces equipped with linear Lie-Poisson brackets. As a result, not only can different non-holonomic systems be compared, but also fairly advanced methods of Poisson geometry and topology can be used for investigating them.
non-holonomic systems, Poisson bracket, Chaplygin ball, Suslov system, Veselova system
Borisov A. V., Mamaev I. S., Tsiganov A. V., Non-holonomic dynamics and Poisson geometry , Russian Mathematical Surveys, 2014, vol. 69, no. 3, pp. 481-538
We discuss an embedding of a vector field for the nonholonomic Routh sphere into a subgroup of commuting Hamiltonian vector fields on six-dimensional phase space. The corresponding Poisson brackets are reduced to the canonical Poisson brackets on the Lie algebra $e^*$(3). It allows us to relate the nonholonomic Routh system with the Hamiltonian system on a cotangent bundle to the sphere with a canonical Poisson structure.
Bizyaev I. A., Tsiganov A. V., On the Routh sphere problem , Journal of Physics A: Mathematical and Theoretical, 2013, vol. 46, no. 8, 085202, 11 pp.
We discuss an embedding of the vector field associated with the nonholonomic Routh sphere in subgroup of the commuting Hamiltonian vector fields associated with this system. We prove that the corresponding Poisson brackets are reduced to canonical ones in the region without of homoclinic trajectories.