Reducing the equations of motion of certain non-holonomic chaplygin systems to lagrangian and hamiltonian form

Moshchuk, N.K.

doi:10.1016/0021-8928(87)90060-8

Cited by 12 publications

(11 citation statements)

References 1 publication

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“…The Poisson bivectorP was obtained in [7] using Chaplygin's reducing multiplier theory. Similar rank-two Poisson structures are discussed in [14,16,21,26].…”

Section: The Chaplygin Hamiltonizationmentioning

confidence: 68%

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Poisson structures for two nonholonomic systems with partially reduced symmetries

Tsiganov¹

2014

Journal of Geometric Mechanics

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We consider nonholonomic systems which symmetry groups consist of two subgroups one of which represents rotations about the axis of symmetry. After nonholonomic reduction by another subgroup the corresponding vector fields on partially reduced phase space are linear combinations of the Hamiltonian and symmetry vector fields. The reduction of the Poisson bivectors associated with the Hamiltonian vector fields to canonical form is discussed.

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“…The Poisson bivectorP was obtained in [7] using Chaplygin's reducing multiplier theory. Similar rank-two Poisson structures are discussed in [14,16,21,26].…”

Section: The Chaplygin Hamiltonizationmentioning

confidence: 68%

“…This construction of rank-two Poisson bivectors for the nonholonomic Stübler model is discussed in [38]. Such rank-two Poisson structures are well-studied [7,14,16,21,26] and, therefore, we will only consider rank-four Poisson structures below.…”

Section: Euler-jacobi Theorem and Rank-two Poisson Structuresmentioning

confidence: 99%

Poisson structures for two nonholonomic systems with partially reduced symmetries

Tsiganov¹

2014

Journal of Geometric Mechanics

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“…where F (γ) is a given function. Notice that (18) has to be fulfilled only on the unit sphere γ 2 = 1. The conditions for solving the equations of this form are well known.…”

Section: Reduction To the E(3)-bracketmentioning

confidence: 99%

Geometrisation of Chaplygin's reducing multiplier theorem

2015

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Abstract. We develop the reducing multiplier theory for a special class of nonholonomic dynamical systems and show that the non-linear Poisson brackets naturally obtained in the framework of this approach are all isomorphic to the LiePoisson e(3)-bracket. As two model examples, we consider the Chaplygin ball problem on the plane and the Veselova system. In particular, we obtain an integrable gyrostatic generalisation of the Veselova system.

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“…Since the dimension of a Lagrangian submanifold is half the dimension of the symplectic leaf, we assume that rank P = 4 almost everywhere. Examples of rank-2 brackets can be found in [37] and [66].…”

Section: The Brackets Corresponding To E(3) and Rank-4 Poisson Structmentioning

confidence: 99%

Non-holonomic dynamics and Poisson geometry

Borisov¹,

Mamaev²,

Tsiganov³

2014

Russ. Math. Surv.

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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.

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Reducing the equations of motion of certain non-holonomic chaplygin systems to lagrangian and hamiltonian form

Cited by 12 publications

References 1 publication

Poisson structures for two nonholonomic systems with partially reduced symmetries

Poisson structures for two nonholonomic systems with partially reduced symmetries

Geometrisation of Chaplygin's reducing multiplier theorem

Non-holonomic dynamics and Poisson geometry

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