Leibniz algebroid associated with a Nambu-Poisson structure

Ibáñez, Raúl; León, Manuel de; Marrero, Juan Carlos; Padrón, Edith

doi:10.1088/0305-4470/32/46/310

Cited by 49 publications

(76 citation statements)

References 29 publications

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“…After computation and in view of the constraints (19) we finally obtain differential system (21). This complete the proof of the theorem.…”

Section: Proof Of Theorems 14 16 and Proposition 19mentioning

confidence: 63%

“…This bracket is known in the literature as the Nambu bracket [29,49,21]. We provide new properties of the Nambu bracket in section 2.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…, g N are arbitrary functions. For more details see [29,49,21]. In this section we show new properties of this bracket which we will use in the proofs and in the applications of the results stated in the two previous sections.…”

Section: Preliminaries and New Properties Of The Nambu Bracketmentioning

confidence: 90%

“…In view of the second Newton law: acceleration is equal to force (see for instance [44]), we obtain that the right hand side of the equations of motion (21) are the generalized forces acting on the mechanical system which depends only on its position. Consequently the field of force F with components…”

Section: Applications To Lagrangian and Hamiltonian Mechanics With Comentioning

confidence: 99%

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Inverse Approach in Ordinary Differential Equations: Applications to Lagrangian and Hamiltonian Mechanics

2014

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Abstract. This paper is on the so called inverse problem of ordinary differential systems, i.e. the problem of determining the differential systems satisfying a set of given properties. More precisely, we characterize under very general assumptions the ordinary differential systems in R N which have a given set of either M ≤ N , or M > N partial integrals, or M < N first integrals, or M ≤ N partial and first integrals. Moreover, for such systems we determine the necessary and sufficient conditions for the existence of N − 1 independent first integrals. For the systems with M < N partial integrals we provide sufficient conditions for the existence of a first integral.We give two relevant applications of the solutions of these inverse problems to constrained Lagrangian and constrained Hamiltonian systems. Additionally we provide a particular solution of the inverse problem in dynamics, and give a generalized solution of the problem of integration of the equation of motion in the classical Suslov problem on SO(3).

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“…After computation and in view of the constraints (19) we finally obtain differential system (21). This complete the proof of the theorem.…”

Section: Proof Of Theorems 14 16 and Proposition 19mentioning

confidence: 63%

“…This bracket is known in the literature as the Nambu bracket [29,49,21]. We provide new properties of the Nambu bracket in section 2.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Section: Preliminaries and New Properties Of The Nambu Bracketmentioning

confidence: 90%

Section: Applications To Lagrangian and Hamiltonian Mechanics With Comentioning

confidence: 99%

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Inverse Approach in Ordinary Differential Equations: Applications to Lagrangian and Hamiltonian Mechanics

2014

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“…Note that ker # Λ = {0} (because the set of regular points of Λ is dense). We can define (see [7]) an R-bilinear operator [[ , ]] :…”

Section: The Choice Of the Cohomology Ifmentioning

confidence: 99%

Computations of Nambu‐Poisson cohomologies

Monnier

2001

International Journal of Mathematics and Mathematical Sciences

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Abstract. We want to associate to an n-vector on a manifold of dimension n a cohomology which generalizes the Poisson cohomology of a 2-dimensional Poisson manifold. Two possibilities are given here. One of them, the Nambu-Poisson cohomology, seems to be the most pertinent. We study these two cohomologies locally, in the case of germs of n-vectors on K n (K = R or C).2000 Mathematics Subject Classification. 53D17. Introduction.A way to study a geometrical object is to associate to it a cohomology. In this paper, we focus on the n-vectors on an n-dimensional manifold M.If n = 2, the 2-vectors on M are the Poisson structures thus, we can consider the Poisson cohomology. In dimension 2, this cohomology has three spaces. The first one, H 0 , is the space of functions whose Hamiltonian vector field is zero (Casimir functions). The second one, H 1 , is the quotient of the space of infinitesimal automorphisms (or Poisson vector fields) by the subspace of Hamiltonian vector fields. The last one, H 2 , describes the deformations of the Poisson structure. In a previous paper (see [9]) we have computed the cohomology of germs at 0 of Poisson structures on K 2 (K = R or C).In order to generalize this cohomology to the n-dimensional case (n ≥ 3), we can follow the same reasoning. These spaces are not necessarily of finite dimension and it is not always easy to describe them precisely.Recently, a team of Spanish researchers has defined a cohomology, called NambuPoisson cohomology, for the Nambu-Poisson structures (see [6]). In this paper, we adapt their construction to our particular case. We will see that this cohomology generalizes in a certain sense the Poisson cohomology in dimension 2. Then we compute locally this cohomology for germs at 0 of n-vectors, with the assumption that f is a quasihomogeneous polynomial of finite codimension ("most of" the germs of n-vectors have this form). This computation is based on a preliminary result that we have shown, in the formal case and in the analytical case (so, the Ꮿ ∞ case is not entirely solved). The techniques we use in this paper are quite the same as in [9].

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