Reduction of generalized Poisson and Nambu-Poisson manifolds
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.
scientific article; zbMATH DE number 1293235
scientific article