Reduction of nonholonomic mechanical systems with symmetries

Cantrijn, Frans; León, Manuel de; Marrero, Juan Carlos; Diego, David Martı́n de

doi:10.1016/s0034-4877(98)80003-7

Cited by 54 publications

(80 citation statements)

References 14 publications

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“…Koiller refers to the more general case, considered in the present paper, as "non-abelian Chaplygin systems". In the literature on nonholonomic systems with symmetry, these systems are also said to be of "principal" or "purely kinematical" type [7,10,20]. Finally, it should be emphasized that the generalized Chaplygin systems studied in [27] are still of a more general type than the ones we consider here in that they are defined on fibre bundles which need not be principal bundles.…”

Section: Nonholonomic Mechanics and Generalized Chaplygin Systemsmentioning

confidence: 99%

“…[10]). This, therefore, in particular applies to the generalized Chaplygin systems considered in this paper.…”

Section: Symplectic Approachmentioning

confidence: 99%

“…Taking into account the available symmetries, we can reduce the number of degrees of freedom of the problem. In the following, we review the various geometric concepts involved in the symplectic approach to this reduction process [10,11,13].…”

Section: Reductionmentioning

confidence: 99%

“…Several concepts and techniques, familiar from the geometric approach to Lagrangian and Hamiltonian mechanics, have been succesfully adapted or extended to the framework of nonholonomic mechanics. Examples of this succesful "transfer of ideas" can be found, among others, in the symplectic and Poisson descriptions of constrained systems [5,15,21,22,26,27,39] and, in particular, in the study of nonholonomic systems with symmetry (reduction and reconstruction of the dynamics, stability of relative equilibria ...) where elements are being used from the theory of symplectic reduction and from the theory of principal connections and Ehresmann connections [5,7,10,13,20,30]. Whereas most of the foregoing treatments are concerned with the autonomous case (i.e.…”

Section: Introductionmentioning

confidence: 99%

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On the geometry of generalized Chaplygin systems

Cantrijn

Cortés

León

et al. 2002

Math. Proc. Camb. Phil. Soc.

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150

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Some aspects of the geometry and the dynamics of generalized Chaplygin systems are investigated. First, two different but complementary approaches to the construction of the reduced dynamics are reviewed: a symplectic approach and an approach based on the theory of affine connections. Both are mutually compared and further completed. Next, a necessary and sufficient condition is derived for the existence of an invariant measure for the reduced dynamics of generalized Chaplygin systems of mechanical type. A simple example is then constructed of a generalized Chaplygin system which does not verify this condition, thereby answering in the negative a question raised by Koiller.

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Section: Nonholonomic Mechanics and Generalized Chaplygin Systemsmentioning

confidence: 99%

“…[10]). This, therefore, in particular applies to the generalized Chaplygin systems considered in this paper.…”

Section: Symplectic Approachmentioning

confidence: 99%

Section: Reductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the geometry of generalized Chaplygin systems

Cantrijn

Cortés

León

et al. 2002

Math. Proc. Camb. Phil. Soc.

Self Cite

150

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“…Parallel to the usual formalism in classical mechanics on the tangent bundle (see e.g. [3]), the constrained system equations of motion can be written in a global form…”

Section: Nonholonomic Systems On a Lie Algebroidmentioning

confidence: 99%

Reduction of Lagrangian mechanics on Lie algebroids

Cariñena

Costa

Santos³

2007

Journal of Geometry and Physics

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We prove that if a surjective submersion which is a homomorphism of Lie algebroids is given, then there exists another homomorphism between the corresponding prolonged Lie algebroids and a relation between the dynamics on these Lie algebroid prolongations is established. We also propose a geometric reduction method for dynamics on Lie algebroids defined by a Lagrangian and the method is applied to regular Lagrangian systems with nonholonomic constraints.

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Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization

Ehlers

Koiller

Montgomery

et al.

Progress in Mathematics

215

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Abstract.A nonholonomic system, for short "NH,'' consists of a configuration space Q n , a Lagrangian L(q,q, t), a nonintegrable constraint distribution H ⊂ T Q, with dynamics governed by Lagrange-d'Alembert's principle. We present here two studies, both using adapted moving frames. In the first we explore the affine connection viewpoint. For natural Lagrangians L = T −V , where we take V = 0 for simplicity, NH-trajectories are geodesics of a (nonmetric) connection ∇ NH which mimics Levi-Civita's. Local geometric invariants are obtained by Cartan's method of equivalence.As an example, we analyze Engel's (2-4) distribution. This is the first such study for a distribution that is not strongly nonholonomic. In the second part we study * The authors thank the Brazilian funding agencies CNPq and

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Reduction of nonholonomic mechanical systems with symmetries

Cited by 54 publications

References 14 publications

On the geometry of generalized Chaplygin systems

On the geometry of generalized Chaplygin systems

Reduction of Lagrangian mechanics on Lie algebroids

Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization

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