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Moment map : 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 \mathfrak be the Lie algebra of ''G'', \mathfrak^
* its dual, and
:\langle, \rangle : \mathfrak^
* \times \mathfrak \to \mathbf
the pairing between the two. Any ξ in \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 \rho(\xi)_x is
:\left.\frac\right|_ \exp(t \xi) \cdot x,
where \exp : \mathfrak \to G is the exponential map and \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 \iota_ \omega \, denote the contraction of this vector field with ω. Because ''G'' acts by symplectomorphisms, it follows that \iota_ \omega \, is closed for all ξ in \mathfrak.
A moment map for the ''G''-action on (''M'', ω) is a map \mu : M \to \mathfrak^
* such that
:d(\langle \mu, \xi \rangle) = \iota_ \omega
for all ξ in \mathfrak. Here \langle \mu, \xi \rangle is the function from ''M'' to R defined by \langle \mu, \xi \rangle(x) = \langle \mu(x), \xi \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 \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 \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)
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