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  "body": "\n## Definition\n\nLet $(M, \\omega)$ be a [[symplectic manifold|CAN490EPRT]] and let $G$ be a Lie group with Lie algebra $\\mathfrak{g}$ and dual space $\\mathfrak{g}^*$.\nLet $G$ [[act|0WCY42EV9V]] smoothly on $M$ by symplectomorphisms.\nFor $\\xi \\in \\mathfrak{g}$, let $\\xi_M$ be the vector field $\\xi_M(x) = \\frac{d}{dt}\\big|_{t=0} \\exp(t\\xi) \\cdot x$.\nA *moment map* for the action is a smooth map $\\mu \\colon M \\to \\mathfrak{g}^*$ such that:\n\n1. For every $\\xi \\in \\mathfrak{g}$, the function $\\mu^\\xi(x) = \\langle \\mu(x), \\xi \\rangle$ satisfies $\\iota_{\\xi_M} \\omega = d\\mu^\\xi$, so $\\xi_M$ is the [[Hamiltonian vector field|C5EH3B4BEP]] of $\\mu^\\xi$.\n2. $\\mu$ is equivariant: $\\mu(g \\cdot x) = \\mathrm{Ad}^*_g \\mu(x)$ for all $g \\in G$ and $x \\in M$, where $\\mathrm{Ad}^*$ is the coadjoint action.\n\nAn action with a moment map is called a *Hamiltonian action*.\n\n## Remarks\n\nExample: let the circle $S^1 = \\R / 2\\pi\\Z$ act on $\\R^2$ by rotations, with $\\omega = dx \\wedge dy$.\nIdentify $\\mathfrak{g}$ and $\\mathfrak{g}^*$ with $\\R$; the generator is $\\xi_M = -y\\,\\partial_x + x\\,\\partial_y$.\nLet $\\mu(x, y) = -\\frac{1}{2}(x^2 + y^2)$.\nThen $\\iota_{\\xi_M} \\omega = -y\\,dy - x\\,dx = d\\mu$, and clause 2 holds because $G$ is abelian and $\\mu$ is rotation invariant.\n\nNon-example: the translations of the torus $T^2 = \\R^2 / \\Z^2$ in the $x$ direction preserve $\\omega = dx \\wedge dy$, but there is no moment map.\nClause 1 would need $\\iota_{\\partial_x} \\omega = dy$ to be the differential of a function on $T^2$, and $dy$ is closed but not exact there.\n\nFor a connected group, clause 2 can be replaced by an infinitesimal condition.\nIt says that $\\xi \\mapsto \\mu^\\xi$ is a Lie algebra homomorphism into functions with the [[Poisson bracket|2S0MXDT4YH]], up to the sign convention.\nWhen $G$ is abelian, a moment map can be changed by adding any constant in $\\mathfrak{g}^*$.\nMany authors use the opposite sign convention in clause 1.\n",
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