Non-holonomic dynamics and Poisson geometry

Abstract

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. Bibliography: 95 titles.

On the Chaplygin Sphere in a Magnetic Field

“…describe the rolling of the Chaplygin sphere on a plane in a potential field [33,11]. By adding an external nonconservative force we have to replace these equations with the equationṡ…”

“…Summing up, we prove the efficiency and relevance of the Dirac method in non-Hamiltonian mechanics using the nonholonomic Chaplygin sphere as an example. In a similar manner we can use compositions of the various known deformations of the Poisson brackets to construct equations of motion for other nonholonomic systems [5,6,9,11,33,34,36].…”

The dynamics of systems with servoconstraints. II

“…This model, which features the dynamics of systems with constraints, is developed in [3, I-V]. In a potential force field, vakonomic motions coincide with the extremals of the variational Lagrange problem with fixed ends: In particular, if V = 0, we obtain the motion along the geodesics of the so-called sub-Riemannian geometry [33,34], which should be distinguished from the nonholonomic geometry developed by Cartan, Vagner and Vranceanu (for details, see [46] and references therein).…”

Bäcklund transformations for the nonholonomic Veselova system

“…This trivial inũ 1,2 andp u1,2 variables HamiltonianĤ =Ĥ 1 has more complicated form in original variables: 9) where ϕ(u) is given by (2.7),…”

“…In nonholonomic mechanics a special attention is given to the systems whose equations of motion after suitable reduction yield conformally Hamiltonian vector field, which can be studied by the standard methods of Hamiltonian mechanics after an appropriate time reparameterization, see reviews [2,6,5,9,15] and the references therein. For instance, because the generic level set of the integrals of motion is independent of time, we can study symmetries of this manifold simultaneously for Hamiltonian and non-Hamiltonian vector fields associated with this level set.…”