翻訳と辞書 Words near each other ・ Poisson (disambiguation) ・ Poisson algebra ・ Poisson binomial distribution ・ Poisson bracket ・ Poisson clumping ・ Poisson d'or ・ Poisson d'or (novel) ・ Poisson distribution ・ Poisson formula ・ Poisson games ・ Poisson hidden Markov model ・ Poisson Hill ・ Poisson image editing ・ Poisson kernel ・ Poisson limit theorem ・ Poisson manifold ・ Poisson number ・ Poisson point process ・ Poisson random measure ・ Poisson regression ・ Poisson ring ・ Poisson sampling ・ Poisson scatter theorem ・ Poisson summation formula ・ Poisson superalgebra ・ Poisson supermanifold ・ Poisson Volant ・ Poisson wavelet ・ Poisson's equation ・ Poisson's ratio Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Poisson manifold ： ウィキペディア英語版 Poisson manifold A Poisson structure on a smooth manifold $M$ is a Lie bracket $\backslash$ (called a Poisson bracket in this special case) on the algebra $=\; f\; \backslash \; +\; g\; \backslash$. Said in another manner, it is a Lie-algebra structure on the vector space of smooth functions on $M$ such that $X\_\; \backslash stackrel\; \backslash :\; \}(M)$ is a vector field for each smooth function $f$, which we call the Hamiltonian vector field associated to $f$. These vector fields span a completely integrable singular foliation, each of whose maximal integral sub-manifolds inherits a symplectic structure. One may thus informally view a Poisson structure on a smooth manifold as a smooth partition of the ambient manifold into even-dimensional symplectic leaves, which are not necessarily of the same dimension. Poisson structures are one instance of Jacobi structures, introduced by André Lichnerowicz in 1977.〔 〕 They were further studied in the classical paper of Alan Weinstein,〔 〕 where many basic structure theorems were first proved, and which exerted a huge influence on the development of Poisson geometry — which today is deeply entangled with non-commutative geometry, integrable systems, topological field theories and representation theory, to name a few. ==Definition== Let $M$ be a smooth manifold. Let $$-bilinear map :$\backslash :\; \}(M)\; \backslash to\; =\; -\; \backslash$. * Jacobi identity: $\backslash \; +\; \backslash \; +\; \backslash \; =\; 0$. * Leibniz's Rule: $\backslash \; =\; f\; \backslash \; +\; g\; \backslash$. The first two conditions ensure that $\backslash$ defines a Lie-algebra structure on $\}(M)$, the adjoint $\backslash :\; \}(M)$ is a derivation of the commutative product on $$. It follows that the bracket $\backslash$ of functions $f$ and $g$ is of the form $\backslash \; =\; \backslash pi(df\; \backslash wedge\; dg)$, where $\backslash pi\; \backslash in\; \backslash Gamma\; \backslash left(\; \backslash bigwedge^\; T\; M\; \backslash right)$ is a smooth bi-vector field. Conversely, given any smooth bi-vector field $\backslash pi$ on $M$, the formula $\backslash \; =\; \backslash pi(df\; \backslash wedge\; dg)$ defines a bilinear skew-symmetric bracket $\backslash$ that automatically obeys Leibniz's rule. The condition that the ensuing $\backslash$ be a Poisson bracket — i.e., satisfy the Jacobi identity — can be characterized by the non-linear partial differential equation $()\; =\; 0$, where :$():\; \}(M)\; \backslash times\; \}(M)\; \backslash to\; \}(M)$ denotes the Schouten–Nijenhuis bracket on multi-vector fields. It is customary and convenient to switch between the bracket and the bi-vector points of view, and we shall do so below. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Poisson manifold」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.