In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, 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 momentum 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 {\displaystyle M} be a manifold with symplectic form ω {\displaystyle \omega } . Suppose that a Lie group G {\displaystyle G} acts on M {\displaystyle M} via symplectomorphisms (that is, the action of each g {\displaystyle g} in G {\displaystyle G} preserves ω {\displaystyle \omega } ). Let g {\displaystyle {\mathfrak {g}}} be the Lie algebra of G {\displaystyle G} , g ∗ {\displaystyle {\mathfrak {g}}^{*}} its dual, and
⟨ ⋅ , ⋅ ⟩ : g ∗ × g → R {\displaystyle \langle \,\cdot ,\cdot \rangle :{\mathfrak {g}}^{*}\times {\mathfrak {g}}\to \mathbb {R} }
the pairing between the two. Any ξ {\displaystyle \xi } in g {\displaystyle {\mathfrak {g}}} induces a vector field ρ ( ξ ) {\displaystyle \rho (\xi )} on M {\displaystyle M} describing the infinitesimal action of ξ {\displaystyle \xi } . To be precise, at a point x {\displaystyle x} in M {\displaystyle M} the vector ρ ( ξ ) x {\displaystyle \rho (\xi )_{x}} is
d d t | t = 0 exp ( t ξ ) ⋅ x , {\displaystyle \left.{\frac {\mathrm {d} }{\mathrm {d} t}}\right|_{t=0}\exp(t\xi )\cdot x,}
where exp : g → G {\displaystyle \exp :{\mathfrak {g}}\to G} is the exponential map and ⋅ {\displaystyle \cdot } denotes the G {\displaystyle G} -action on M {\displaystyle M} . Let ι ρ ( ξ ) ω {\displaystyle \iota _{\rho (\xi )}\omega \,} denote the contraction of this vector field with ω {\displaystyle \omega } . Because G {\displaystyle G} acts by symplectomorphisms, it follows, by Cartan’s Magic Formula, that ι ρ ( ξ ) ω {\displaystyle \iota _{\rho (\xi )}\omega \,} is closed (for all ξ {\displaystyle \xi } in g {\displaystyle {\mathfrak {g}}} ). Suppose that ι ρ ( ξ ) ω {\displaystyle \iota _{\rho (\xi )}\omega \,} is not just closed but also exact, so that ι ρ ( ξ ) ω = d H ξ {\displaystyle \iota _{\rho (\xi )}\omega =\mathrm {d} H_{\xi }} for some function H ξ : M → R {\displaystyle H_{\xi }:M\to \mathbb {R} } . If this holds, then one may choose the H ξ {\displaystyle H_{\xi }} to make the map ξ ↦ H ξ {\displaystyle \xi \mapsto H_{\xi }} linear. A momentum map for the G {\displaystyle G} -action on ( M , ω ) {\displaystyle (M,\omega )} is a map μ : M → g ∗ {\displaystyle \mu :M\to {\mathfrak {g}}^{*}} such that
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