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  "body": "\n## Summary\n\nA moment map, also called a momentum map, packages the conserved quantities of a symmetry of a [[symplectic manifold|CAN490EPRT]] into one map.\nWhen a Lie group $G$ acts on $(M, \\omega)$, a moment map is a map $\\mu \\colon M \\to \\mathfrak{g}^*$ whose components generate the action as [[Hamiltonian vector fields|C5EH3B4BEP]].\nIt is the main tool for symplectic reduction, for convexity theorems and for toric geometry.\n\n## Explanations\n\n### L1 human\n\nIn mechanics, a symmetry gives a conserved quantity: translations give linear momentum and rotations give angular momentum.\nThe moment map collects these quantities into a single map from phase space to the dual of the Lie algebra of the symmetry group.\nFor the circle rotating the plane about the origin, the moment map is, up to sign and an added constant, half the squared distance to the origin.\nIts level sets are the circles that the rotation preserves.\n",
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