Nonholonomic constraints: A new viewpoint

Grabowski, Janusz; León, Manuel de; Marrero, Juan Carlos; Diego, David Martı́n de

doi:10.1063/1.3049752

Cited by 65 publications

(77 citation statements)

References 29 publications

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Section: The Gyroscopic Tensormentioning

confidence: 98%

“…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 .…”

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confidence: 98%

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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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Section: The Gyroscopic Tensormentioning

confidence: 98%

mentioning

confidence: 98%

The geometry of nonholonomic Chaplygin systems revisited

García-Naranjo

Marrero

2020

Nonlinearity

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“…The use of Lie algebroids or even more general concepts leads us to develop a new and very general setting for hamiltonian mechanics [37,58,84].…”

Section: Generalized Hamiltonian Systemsmentioning

confidence: 99%

A historical review on nonholomic mechanics

León

2011

RACSAM

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The aim of this paper is to present a short historical/scientific review on nonholonomic mechanics, with special emphasis on the latest developments. Indeed, the use of differential geometric tools has permitted in the last 25 years a fast and unsuspected advance in the theory, particularly in a better understanding of symmetries and reduction, Hamilton-Jacobi theory and integrability characterizations, and the construction of suitable geometric integrators. The last part of the paper is devoted to discuss the latest results in Hamilton-Jacobi theory for nonholonomic dynamics using our own approach.

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“…According also to [3,5,7], the interest in almost Lie algebroids comes also from that they are involved in nonholonomic geometry.…”

Section: Introductionmentioning

confidence: 99%

Almost Lie Algebroids and Characteristic Classes

Popescu¹,

Popescu²

2019

SIGMA

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Almost Lie algebroids are generalizations of Lie algebroids, when the Jacobiator is not necessary null. A simple example is given, for which a Lie algebroid bracket or a Courant bundle is not possible for the given anchor, but a natural extension of the bundle and the new anchor allows a Lie algebroid bracket. A cohomology and related characteristic classes of an almost Lie algebroid are also constructed. We prove that these characteristic classes are all pull-backs of the characteristic classes of the base space, as in the case of a Lie algebroid.

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Nonholonomic constraints: A new viewpoint

Cited by 65 publications

References 29 publications

The geometry of nonholonomic Chaplygin systems revisited

The geometry of nonholonomic Chaplygin systems revisited

A historical review on nonholomic mechanics

Almost Lie Algebroids and Characteristic Classes

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