Reduction of generalized Poisson and Nambu-Poisson manifolds
type: Item
instance of: scholarly article
title: Reduction of generalized Poisson and Nambu-Poisson manifolds
zbMATH Open document ID: 0931.37024
DOI: 10.1016/S0034-4877(98)80005-0
author: Raúl Ibañez
author: Juan Carlos Marrero
author: David Martín de Diego
author: Manuel de León
published in: Reports on Mathematical Physics
publication date: 2000-02-27T00:00:00Z
review text: For Poisson manifolds the most general reduction procedure is that developed by \textit{J. Marsden} and \textit{T. Ratiu} [Lett. Math. Phys. 11, 161-169 (1986; Zbl 0602.58016)]. In this paper, this result is extended to Nambu-Poisson and generalized Poisson manifolds. See \textit{R. Ibañez, M. de León, J.C. Marrero} and the reviewer [J. Math. Phys. 38, 2332-2344 (1997; Zbl 0878.58024)] for more details about extensions of Poisson brackets to \(n\)-brackets. Consider \((M,\Lambda)\) a Nambu-Poisson manifold, (resp., generalized Poisson manifold), a submanifold \(N\) and a vector subbundle \(E\) of \(TM_{|N}\) satisfying: \(E\cap TN\) is an integrable subbundle of \(TN\) (so it defines a foliation \({\mathcal F}\) on \(N\)); the foliation \({\mathcal F}\) is regular, then, the space of leaves \(N/{\mathcal F}\) is a manifold and the projection \(\pi:N\rightarrow N/{\mathcal F}\) is a submersion; and, finally, if \(\{ ,\dots , \}_M\) is the Nambu-Poisson (resp., generalized Poisson) bracket, the bundle \(E\) leaves the bracket invariant. Under these conditions, the triple \((M,N,E)\) is Nambu-Poisson (resp. generalized Poisson) reductive if \(N/{\mathcal F}\) has a Nambu-Poisson (resp., generalized Poisson) structure such that for functions \(f_1,\dots f_n \) on \(N/{\mathcal F}\) \[ \{f_1,\dots ,f_n\}_{N/{\mathcal F}}\circ \pi F_1,\dots ,F_n\}_{|M}\circ i, \] where \(i:N\rightarrow M\) is the canonical inclusion, \(\{,\dots ,\}_{N/{\mathcal F}}\) is the bracket on \(N/{\mathcal F}\) and \(F_j\) is a smooth extension of \(f_j\circ \pi\), with \(dF_j\) vanishing on \(E\). The authors prove the following main result: \((M,N,E)\) is Nambu-Poisson (resp., generalized Poisson) reductive if and only if: \[ \#(E^0)\subseteq TN+E. \] Finally, the relationship between constrained Hamiltonian systems and Nambu-Poisson brackets is studied and it is compared the reduction of the Nambu-Poisson bracket with the reduction of the associated constrained Hamiltonian system.
reviewed by: David Martín de Diego
Mathematics Subject Classification ID: 37K05
Mathematics Subject Classification ID: 37K30
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 70G45
zbMATH DE Number: 1293235
zbMATH Keywords: Marsden-Ratiu reduction
zbMATH Keywords: Nambu-Poisson manifolds
zbMATH Keywords: constrained Hamiltonian systems
zbMATH Keywords: Poisson manifolds
zbMATH Keywords: generalized Poisson manifolds
MaRDI profile type: Publication
cites work: On the higher-order generalizations of Poisson structures
cites work: New generalized Poisson structures
cites work: The Schouten - Nijenhuis bracket, cohomology and generalized Poisson structures
cites work: Geometric Lagrangian approach to first-order systems and applications
cites work: Dynamical symmetries and Nambu mechanics
cites work: Generalization of Nambu's mechanics
cites work: On Nambu's generalized Hamiltonian mechanics
cites work: Leibniz and Lie algebra structures for Nambu algebra
cites work: Foundations of Quantum Mechanics
cites work: Q3529080
cites work: Some remarks concerning Nambu mechanics
cites work: Q3049554
cites work: Q3900713
cites work: Nambu tensors and commuting vector fields
cites work: Dynamics of generalized Poisson and Nambu–Poisson brackets
cites work: Q3819195
cites work: Constrained systems described by Nambu mechanics
cites work: Q4400543
cites work: Reduction of Poisson manifolds
cites work: Generalized Hamiltonian Dynamics
cites work: Hamiltonian dynamics in symplectic phase space and Nambu’s formulation of mechanics
cites work: Q3719916
cites work: On generalized Nambu mechanics
cites work: On foundation of the generalized Nambu mechanics
cites work: Errata and addenda
full work available at URL: s0034-4877(98)80005-0
OpenAlex ID: W2086458961
description: scientific article; zbMATH DE number 1293235