Lagrangian submanifolds and dynamics on Lie algebroids

León, Manuel de; Marrero, Juan Carlos; Martínez, Eduardo

doi:10.1088/0305-4470/38/24/r01

Cited by 154 publications

(415 citation statements)

References 31 publications

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“…It is easy to prove that Ω is nondegenerate and closed, that is, it is a symplectic section of T EΓ E * Γ (see [23]). Now, if Z is a section of τ : E Γ → M then there is a unique vector field Z * c on E * Γ , the complete lift of X to E * Γ , satisfying the two following conditions:…”

Section: Lie Groupoidsmentioning

confidence: 99%

“…for X ∈ Sec(τ ) (see [23]). Here, if X is a section of τ : E Γ → M then X is the linear function X ∈ C ∞ (E * ) defined by X(a * ) = a * (X(τ * (a * ))), for all a * ∈ E * .…”

Section: Lie Groupoidsmentioning

confidence: 99%

“…In other words, 10) where d is the differential of the Lie algebroid τ τ * : T EΓ E * Γ → E * Γ (for more details, see [23]). …”

Section: Lie Groupoidsmentioning

confidence: 99%

“…By using the Lagrange multiplier rule, we obtain that with the conditionq(t) ∈ M,λ a being the Lagrange multipliers to be determined. Recently, J. Cortés et al [9] (see also [11,38,39]) proposed a unified framework for nonholonomic systems in the Lie algebroid setting that we will use along this paper generalizing some previous work for free Lagrangian mechanics on Lie algebroids (see, for instance, [23,33,34,35]). …”

Section: Introductionmentioning

confidence: 99%

“…is the E-tangent bundle to P or the prolongation of E over the fibration π : P → M (for more details, see [23]). …”

Section: Introductionmentioning

confidence: 99%

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Discrete Nonholonomic Lagrangian Systems on Lie Groupoids

et al. 2007

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Abstract. This paper studies the construction of geometric integrators for nonholonomic systems. We derive the nonholonomic discrete Euler-Lagrange equations in a setting which permits to deduce geometric integrators for continuous nonholonomic systems (reduced or not). The formalism is given in terms of Lie groupoids, specifying a discrete Lagrangian and a constraint submanifold on it. Additionally, it is necessary to fix a vector subbundle of the Lie algebroid associated to the Lie groupoid. We also discuss the existence of nonholonomic evolution operators in terms of the discrete nonholonomic Legendre transformations and in terms of adequate decompositions of the prolongation of the Lie groupoid. The characterization of the reversibility of the evolution operator and the discrete nonholonomic momentum equation are also considered. Finally, we illustrate with several classical examples the wide range of application of the theory (the discrete nonholonomic constrained particle, the Suslov system, the Chaplygin sleigh, the Veselova system, the rolling ball on a rotating table and the two wheeled planar mobile robot).

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Section: Lie Groupoidsmentioning

confidence: 99%

“…for X ∈ Sec(τ ) (see [23]). Here, if X is a section of τ : E Γ → M then X is the linear function X ∈ C ∞ (E * ) defined by X(a * ) = a * (X(τ * (a * ))), for all a * ∈ E * .…”

Section: Lie Groupoidsmentioning

confidence: 99%

“…In other words, 10) where d is the differential of the Lie algebroid τ τ * : T EΓ E * Γ → E * Γ (for more details, see [23]). …”

Section: Lie Groupoidsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…is the E-tangent bundle to P or the prolongation of E over the fibration π : P → M (for more details, see [23]). …”

Section: Introductionmentioning

confidence: 99%

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Discrete Nonholonomic Lagrangian Systems on Lie Groupoids

et al. 2007

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Left‐symmetric algebroids

Liu

Sheng

Bai

et al. 2016

Mathematische Nachrichten

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In this paper, we introduce a notion of a left-symmetric algebroid, which is a generalization of a left-symmetric algebra from a vector space to a vector bundle. The left multiplication gives rise to a representation of the corresponding sub-adjacent Lie algebroid. We construct left-symmetric algebroids from O-operators on Lie algebroids. We study phase spaces of Lie algebroids in terms of left-symmetric algebroids. Representations of left-symmetric algebroids are studied in detail. At last, we study deformations of left-symmetric algebroids, which could be controlled by the second cohomology class in the deformation cohomology.

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The Local Description of Discrete Mechanics

Marrero

Diego

Martínez

2015

Geometry, Mechanics, and Dynamics

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In this paper, we introduce local expressions for discrete Mechanics. To apply our results simultaneously to several interesting cases, we derive these local expressions in the framework of Lie groupoids, following the program proposed by Alan Weinstein in [19]. To do this, we will need some results on the geometry of Lie groupoids, as, for instance, the construction of symmetric neighborhoods or the existence of local bisections. These local descriptions will be particular useful for the explicit construction of geometric integrators for mechanical systems (reduced or not), in particular, discrete Euler-Lagrange equations, discrete Euler-Poincaré equations, discrete Lagrange-Poincaré equations... The results contained in this paper can be considered as a local version of the study that we have started in [13], on the geometry of discrete Mechanics on Lie groupoids. Dedicated to the memory of J. E. MarsdenMathematics Subject Classification (2010): 17B66, 22A22, 70G45, 70Hxx.

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Lagrangian submanifolds and dynamics on Lie algebroids

Cited by 154 publications

References 31 publications

Discrete Nonholonomic Lagrangian Systems on Lie Groupoids

Discrete Nonholonomic Lagrangian Systems on Lie Groupoids

Left‐symmetric algebroids

The Local Description of Discrete Mechanics

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