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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Marsden-Weinstein quotient ： ウィキペディア英語版 Moment map In mathematics, specifically in symplectic geometry, the momentum map (or moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The moment map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic (Marsden–Weinstein) quotients, discussed below, and symplectic cuts and sums. == Formal definition == Let ''M'' be a manifold with symplectic form ω. Suppose that a Lie group ''G'' acts on ''M'' via symplectomorphisms (that is, the action of each ''g'' in ''G'' preserves ω). Let $\backslash mathfrak$ be the Lie algebra of ''G'', $\backslash mathfrak^$ * its dual, and :$\backslash langle,\; \backslash rangle\; :\; \backslash mathfrak^$ * \times \mathfrak \to \mathbf the pairing between the two. Any ξ in $\backslash mathfrak$ induces a vector field ρ(ξ) on ''M'' describing the infinitesimal action of ξ. To be precise, at a point ''x'' in ''M'' the vector $\backslash rho(\backslash xi)\_x$ is :$\backslash left.\backslash frac\backslash right|\_\; \backslash exp(t\; \backslash xi)\; \backslash cdot\; x,$ where $\backslash exp\; :\; \backslash mathfrak\; \backslash to\; G$ is the exponential map and $\backslash cdot$ denotes the ''G''-action on ''M''.〔The vector field ρ(ξ) is called sometimes the Killing vector field relative to the action of the one-parameter subgroup generated by ξ. See, for instance, 〕 Let $\backslash iota\_\; \backslash omega\; \backslash ,$ denote the contraction of this vector field with ω. Because ''G'' acts by symplectomorphisms, it follows that $\backslash iota\_\; \backslash omega\; \backslash ,$ is closed for all ξ in $\backslash mathfrak$. A moment map for the ''G''-action on (''M'', ω) is a map $\backslash mu\; :\; M\; \backslash to\; \backslash mathfrak^$ * such that :$d(\backslash langle\; \backslash mu,\; \backslash xi\; \backslash rangle)\; =\; \backslash iota\_\; \backslash omega$ for all ξ in $\backslash mathfrak$. Here $\backslash langle\; \backslash mu,\; \backslash xi\; \backslash rangle$ is the function from ''M'' to R defined by $\backslash langle\; \backslash mu,\; \backslash xi\; \backslash rangle(x)\; =\; \backslash langle\; \backslash mu(x),\; \backslash xi\; \backslash rangle$. The moment map is uniquely defined up to an additive constant of integration. A moment map is often also required to be ''G''-equivariant, where ''G'' acts on $\backslash mathfrak^$ * via the coadjoint action. If the group is compact or semisimple, then the constant of integration can always be chosen to make the moment map coadjoint equivariant; however in general the coadjoint action must be modified to make the map equivariant (this is the case for example for the Euclidean group). The modification is by a 1-cocycle on the group with values in $\backslash mathfrak^$ *, as first described by Souriau (1970). 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「Moment map」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.