Gauge Transformations, Twisted Poisson Brackets and Hamiltonization of Nonholonomic Systems

Balseiro, Paula; García-Naranjo, Luis C.

doi:10.1007/s00205-012-0512-9

Cited by 43 publications

(143 citation statements)

References 26 publications

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“…1 The gyroscopic tensor actually coincides, up to a sign, with the tensor field C considered by Koiller [28,Proposition 8.5] and Cantrijn et al [11, Page 337] (see [21]). 2 The form of the equations (2.2) and (2.7) indicates that there is an interesting connection between the gyroscopic coefficients and the structure coefficients of other sophisticated geometric frameworks that have been developed to formulate the equations of motion of nonholonomic systems [23,30]. Theorem 2.1 (Cantrijn et al [11]).…”

Section: Invariant Measures For Chaplygin Systemsmentioning

confidence: 99%

“…A substantial amount of research in nonholonomic mechanics in recent years has focused on Hamiltonisation (see e.g. [7,13,15,14,8,16,26,20,5,2,6] and the references therein). Roughly speaking this is the process by which, via symmetry reduction and a time reparametrisation, the equations of motion of certain nonholonomic systems take a Hamiltonian form.…”

Section: Introductionmentioning

confidence: 99%

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Generalisation of Chaplygin’s reducing multiplier theorem with an application to multi-dimensional nonholonomic dynamics

García-Naranjo¹

2019

J. Phys. A: Math. Theor.

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A generalisation of Chaplygin's Reducing Multiplier Theorem is given by providing sufficient conditions for the Hamiltonisation of Chaplygin nonholonomic systems with an arbitrary number r of degrees of freedom via Chaplygin's multiplier method. The crucial point in the construction is to add an hypothesis of geometric nature that controls the interplay between the kinetic energy metric and the non-integrability of the constraint distribution. Such hypothesis can be systematically examined in concrete examples, and is automatically satisfied in the case r = 2 encountered in the original formulation of Chaplygin's theorem. Our results are applied to prove the Hamiltonisation of a multi-dimensional generalisation of the problem of a symmetric rigid body with a flat face that rolls without slipping or spinning over a sphere.

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Section: Invariant Measures For Chaplygin Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Generalisation of Chaplygin’s reducing multiplier theorem with an application to multi-dimensional nonholonomic dynamics

García-Naranjo¹

2019

J. Phys. A: Math. Theor.

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“…In particular, it is of great interest to determine geometric conditions, usually related with symmetry, that lead to the existence of invariants of the flow, such as first integrals, volume forms and Poisson or symplectic structures (see e.g. [36,6,11,52,19,13,15,2,9,21] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

The geometry of nonholonomic Chaplygin systems revisited

García-Naranjo

Marrero

2020

Nonlinearity

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We consider nonholonomic Chaplygin systems and associate to them a (1,2) tensor field on the shape space, that we term the gyroscopic tensor, and that measures the interplay between the non-integrability of the constraint distribution and the kinetic energy metric. We show how this tensor may be naturally used to derive an almost symplectic description of the reduced dynamics. Moreover, we express sufficient conditions for measure preservation and Hamiltonisation via Chaplygin's reducing multiplier method in terms of the properties of this tensor. The theory is used to give a new proof of the remarkable Hamiltonisation of the multi-dimensional Veselova system obtained by Fedorov and Jovanović in [19,20]. where g denotes the Lie algebra of G and g ⋅ q the tangent space to the G-orbit through q.These systems were first considered by Chaplygin and Hamel around the year 1900, and their geometric features have since been investigated by a number of authors e.g. [1,30,47,36,6,11,19,13,10,27,3,21] and references therein. Their key feature is that the reduced equations of motion may be formulated as an unconstrained, forced mechanical system on the shape space S = Q G. The dimension r of S is termed the number of degrees of freedom and, because of (1.1), it coincides with the rank of the constraint distribution D (in particular, the non-integrability of D implies that r ≥ 2).Our contribution to the subject is to highlight the relevance of a tensor field T defined on S, that we term the gyroscopic tensor, in the structure of the reduced equations of motion and their properties. Moreover, using this tensor, we are able to single out a very special class of systems, that we term φ -simple, which possess an invariant measure and allow a Hamiltonisation, and for which there exist non-trivial examples. The gyroscopic tensorThe gyroscopic tensor T is introduced in Definition 3.3. It is a (1,2) skew-symmetric tensor field on the shape space S, that to a pair of vector fields Y,Z on S, assigns a third vector field T (Y,Z) on S. The assignment is done in a manner that measures the interplay between the nonintegrability of the noholonomic constraint distribution and the kinetic energy of the system. In particular, in the case of holonomic constraints, where the constraint distribution is integrable, we have T = 0. We mention that there is a close relation between T and the geometric formulation of nonholonomic systems in terms of linear almost Poisson brackets on vector bundles [26,40] (see also [21]).Although the tensor T appears in the previous works of Koiller [36] and Cantrijn et al. [11] (with an alternative definition than the one that we present here), its dynamical relevance had not been fully appreciated until the recent work García-Naranjo [24] where sufficient conditions for Hamiltonisation were given in terms of the coordinate representation of T . This work continues the research started in [24] by providing a coordinate-free definition of the gyroscopic tensor (Definition 3.3), and studying in depth its role in the al...

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“…This paper explores the connection between the presence of conserved quantities for a nonholonomic system and its hamiltonization, as raised in [31]. Using the geometric techniques developed in [1,3], we show that, for certain types of symmetries admitting conserved quantities, one can distinguish particular 2-forms that can be used to modify the classical nonholonomic bracket (by means of gauge transformations); the reduction of such modified brackets to the orbit space are genuine Poisson brackets, relative to which the reduced equations of motion are hamiltonian. We show that all conditions for this procedure to work are met for a concrete set of examples, namely solids of revolution rolling on a plane without sliding as well as the classical example of an inhomogeneous ball rolling on a plane.…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

Hamiltonization of Solids of Revolution Through Reduction

Balseiro

2017

J Nonlinear Sci

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In this paper we study the relation between conserved quantities of nonholonomic systems and the hamiltonization problem employing the geometric methods of [1,3]. We illustrate the theory with classical examples describing the dynamics of solids of revolution rolling without sliding on a plane. In these cases, using the existence of two conserved quantities we obtain, by means of gauge transformations and symmetry reduction, genuine Poisson brackets describing the reduced dynamics.

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Gauge Transformations, Twisted Poisson Brackets and Hamiltonization of Nonholonomic Systems

Cited by 43 publications

References 26 publications

Generalisation of Chaplygin’s reducing multiplier theorem with an application to multi-dimensional nonholonomic dynamics

Generalisation of Chaplygin’s reducing multiplier theorem with an application to multi-dimensional nonholonomic dynamics

The geometry of nonholonomic Chaplygin systems revisited

Hamiltonization of Solids of Revolution Through Reduction

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