Infinitesimal Poisson algebras and linearization of Hamiltonian systems
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Infinitesimal Poisson algebras and linearization of Hamiltonian systems J. C. Ruíz-Pantaleón1
· D. García-Beltrán2
· Yu. Vorobiev3
Received: 17 April 2020 / Accepted: 24 August 2020 © Springer Nature B.V. 2020
Abstract Using the notion of a contravariant derivative, we give some algebraic and geometric characterizations of Poisson algebras associated to the infinitesimal data of Poisson submanifolds. We show that such a class of Poisson algebras provides a suitable framework for the study of the Hamiltonization problem for the linearized dynamics along Poisson submanifolds. Keywords Poisson algebra · Poisson submanifold · Hamiltonian system · Linearization · Contravariant derivative Mathematics Subject Classification 53D17 · 37J05 · 53C05
1 Introduction In this paper, we describe a class of Poisson algebras which appear in the context of infinitesimal geometry of Poisson submanifolds, known also as first-class constraints [13,21,22]. One of our motivations is to provide a suitable framework for a nonintrinsic Hamiltonian formulation of linearized Hamiltonian dynamics along Poisson submanifolds of nonzero dimension. This question can be viewed as a part of a general Hamiltonization problem for projectable dynamics on fibered manifolds studied in various situations in [2,14,18–20]. The main feature of our case is that we have to state the Hamiltonization problem in a class of Poisson algebras which do not define any Poisson structures, in general. This situation is related to the problem of the construction of first-order approximations of Poisson structures around
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J. C. Ruíz-Pantaleón [email protected] D. García-Beltrán [email protected] Yu. Vorobiev [email protected]
1
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico City, Mexico
2
CONACyT Research-Fellow, Departamento de Matemáticas, Universidad de Sonora, Hermosillo, Mexico
3
Departamento de Matemáticas, Universidad de Sonora, Hermosillo, Mexico
123
Annals of Global Analysis and Geometry
Poisson submanifolds [11,12] which is only well-studied in the case of symplectic leaves [17,18]. Let S be an embedded Poisson submanifold of a Poisson manifold (M, {, } M ). Then, for every H ∈ C∞ M , the Hamiltonian vector field X H on M is tangent to S and hence can be linearized along S. The linearized procedure for X H at S leads to a linear vector field var S X H ∈ X ¯ lin (E) on the normal bundle of S defined as a quotient vector bundle E = T S M/TS. In the zero-dimensional case, when S = {q} is a singular point of the Poisson structure on M, the linear vector field var S X H is Hamiltonian relative to the induced Lie– Poisson bracket on E = Tq M. If dim S > 0, then the linearized dynamical model associated with var S X H , called a first variation system, does not inherit any natural Hamiltonian structure from the original Hamiltonian system. This fact gives rise to the so-called Hamiltonization problem for var S X H which is formulated in a class of Poisson algebras on the space of fiberwise affine functions C∞ aff
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