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| In [[mathematics]], specifically in [[symplectic geometry]], the '''momentum map''' (or '''moment map''') is a tool associated with a [[Hamiltonian action|Hamiltonian]] [[group action|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 cut]]s and [[symplectic sum|sums]].
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| == Formal definition ==
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| Let ''M'' be a manifold with [[symplectic form]] ω. Suppose that a Lie group ''G'' acts on ''M'' via [[symplectomorphism]]s (that is, the action of each ''g'' in ''G'' preserves ω). Let <math>\mathfrak{g}</math> be the [[Lie algebra]] of ''G'', <math>\mathfrak{g}^*</math> its [[dual space|dual]], and
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| :<math>\langle, \rangle : \mathfrak{g}^* \times \mathfrak{g} \to \mathbf{R}</math>
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| the pairing between the two. Any ξ in <math>\mathfrak{g}</math> 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 <math>\rho(\xi)_x</math> is
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| :<math>\left.\frac{d}{dt}\right|_{t = 0} \exp(t \xi) \cdot x,</math>
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| where <math>\exp : \mathfrak{g} \to G</math> is the [[exponential map]] and <math>\cdot</math> denotes the ''G''-action on ''M''.<ref>The vector field ρ(ξ) is called sometimes the [[Killing vector field#Generalizations|Killing vector field]] relative to the action of the [[Exponential map#Definitions|one-parameter subgroup]] generated by ξ. See, for instance, {{harv|Choquet-Bruhat|DeWitt-Morette|1977}}</ref> Let <math>\iota_{\rho(\xi)} \omega \,</math> denote the [[Interior product|contraction]] of this vector field with ω. Because ''G'' acts by symplectomorphisms, it follows that <math>\iota_{\rho(\xi)} \omega \,</math> is [[closed and exact differential forms|closed]] for all ξ in <math>\mathfrak{g}</math>.
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| A '''moment map''' for the ''G''-action on (''M'', ω) is a map <math>\mu : M \to \mathfrak{g}^*</math> such that
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| :<math>d(\langle \mu, \xi \rangle) = \iota_{\rho(\xi)} \omega</math>
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| for all ξ in <math>\mathfrak{g}</math>. Here <math>\langle \mu, \xi \rangle</math> is the function from ''M'' to '''R''' defined by <math>\langle \mu, \xi \rangle(x) = \langle \mu(x), \xi \rangle</math>. The moment map is uniquely defined up to an additive constant of integration.
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| A moment map is often also required to be ''G''-equivariant, where ''G'' acts on <math>\mathfrak{g}^*</math> 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-[[Group cohomology|cocycle]] on the group with values in <math>\mathfrak{g}^*</math>, as first described by Souriau (1970).
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| == Hamiltonian group actions ==
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| The definition of the moment map requires <math>\iota_{\rho(\xi)} \omega</math> to be [[closed and exact differential forms|exact]]. In practice it is useful to make an even stronger assumption. The ''G''-action is said to be '''Hamiltonian''' if and only if the following conditions hold. First, for every ξ in <math>\mathfrak{g}</math> the one-form <math>\iota_{\rho(\xi)} \omega</math> is exact, meaning that it equals <math>dH_\xi</math> for some smooth function
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| :<math>H_\xi : M \to \mathbf{R}.</math>
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| If this holds, then one may choose the <math>H_\xi</math> to make the map <math>\xi \mapsto H_\xi</math> linear. The second requirement for the ''G''-action to be Hamiltonian is that the map <math>\xi \mapsto H_\xi</math> be a Lie algebra homomorphism from <math>\mathfrak{g}</math> to the algebra of smooth functions on ''M'' under the [[Poisson bracket]].
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| If the action of ''G'' on (''M'', ω) is Hamiltonian in this sense, then a moment map is a map <math>\mu : M\to \mathfrak{g}^*</math> such that writing <math>H_\xi = \langle \mu, \xi \rangle</math> defines a Lie algebra homomorphism <math>\xi \mapsto H_\xi</math> satisfying <math>\rho(\xi) = X_{H_\xi}</math>. Here <math>X_{H_\xi}</math> is the vector field of the Hamiltonian <math>H_\xi</math>, defined by
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| :<math>\iota_{X_{H_\xi}} \omega = d H_\xi.</math>
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| == Examples ==
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| In the case of a Hamiltonian action of the circle ''G'' = U(1), the Lie algebra dual <math>\mathfrak{g}^*</math> is naturally identified with '''R''', and the moment map is simply the Hamiltonian function that generates the circle action.
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| Another classical case occurs when ''M'' is the [[cotangent bundle]] of '''R'''<sup>3</sup> and ''G'' is the [[Euclidean group]] generated by rotations and translations. That is, ''G'' is a six-dimensional group, the [[semidirect product]] of SO(3) and '''R'''<sup>3</sup>. The six components of the moment map are then the three angular momenta and the three linear momenta.
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| == Symplectic quotients ==
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| Suppose that the action of a [[compact Lie group]] ''G'' on the symplectic manifold (''M'', ω) is Hamiltonian, as defined above, with moment map <math>\mu : M\to \mathfrak{g}^*</math>. From the Hamiltonian condition it follows that <math>\mu^{-1}(0)</math> is invariant under ''G''.
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| Assume now that 0 is a regular value of μ and that ''G'' acts freely and properly on <math>\mu^{-1}(0)</math>. Thus <math>\mu^{-1}(0)</math> and its [[quotient space|quotient]] <math>\mu^{-1}(0) / G</math> are both manifolds. The quotient inherits a symplectic form from ''M''; that is, there is a unique symplectic form on the quotient whose [[pullback (differential geometry)|pullback]] to <math>\mu^{-1}(0)</math> equals the restriction of ω to <math>\mu^{-1}(0)</math>. Thus the quotient is a symplectic manifold, called the '''Marsden–Weinstein quotient''', '''symplectic quotient''' or '''symplectic reduction''' of ''M'' by ''G'' and is denoted <math>M/\!\!/G</math>. Its dimension equals the dimension of ''M'' minus twice the dimension of ''G''.
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| ==See also==
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| * [[Poisson-Lie group]]
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| * [[Toric manifold]]
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| * [[Geometric Mechanics]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * J.-M. Souriau, ''Structure des systèmes dynamiques'', Maîtrises de mathématiques, Dunod, Paris, 1970. ISSN 0750-2435.
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| * [[S. K. Donaldson]] and P. B. Kronheimer, ''The Geometry of Four-Manifolds'', Oxford Science Publications, 1990. ISBN 0-19-850269-9.
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| * [[Dusa McDuff]] and Dietmar Salamon, ''Introduction to Symplectic Topology'', Oxford Science Publications, 1998. ISBN 0-19-850451-9.
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| *{{Citation
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| |last = Choquet-Bruhat
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| |first = Yvonne
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| |authorlink = Yvonne Choquet-Bruhat
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| |first2 = Cécile |last2=DeWitt-Morette| title = Analysis, Manifolds and Physics| publisher = Elsevier| year= 1977| location = Amsterdam |ISBN = 978-0-7204-0494-4}}
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| * {{cite book| last1=Ortega| first1=Juan-Pablo|last2=Ratiu| first2=Tudor S.| title=Momentum maps and Hamiltonian reduction|publisher = Birkhauser Boston|series=Progress in Mathematics|volume = 222|year = 2004|isbn = 0-8176-4307-9}}
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| {{DEFAULTSORT:Moment Map}}
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| [[Category:Symplectic geometry]]
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| [[Category:Hamiltonian mechanics]]
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| [[Category:Group actions]]
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