---
tag: ZARM34FZ6Z
kind: concept
name: Moment map
msc: [53D20, 37J39]
sources:
  - url: https://en.wikipedia.org/wiki/Momentum_map
same_as:
  wikidata: Q6897118
  wikipedia: Momentum map
---

## Summary

A moment map, also called a momentum map, packages the conserved quantities of a symmetry of a [[symplectic manifold|CAN490EPRT]] into one map.
When 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]].
It is the main tool for symplectic reduction, for convexity theorems and for toric geometry.

## Explanations

### L1 human

In mechanics, a symmetry gives a conserved quantity: translations give linear momentum and rotations give angular momentum.
The moment map collects these quantities into a single map from phase space to the dual of the Lie algebra of the symmetry group.
For 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.
Its level sets are the circles that the rotation preserves.