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.

Let ${\displaystyle M}$ be a manifold with symplectic form ${\displaystyle \omega }$. Suppose that a Lie group ${\displaystyle G}$ acts on ${\displaystyle M}$ via symplectomorphisms (that is, the action of each ${\displaystyle g}$ in ${\displaystyle G}$ preserves ${\displaystyle \omega }$). Let ${\displaystyle {\mathfrak {g}}}$ be the Lie algebra of ${\displaystyle G}$, ${\displaystyle {\mathfrak {g}}^{*}}$ its dual, and

the pairing between the two. Any ${\displaystyle \xi }$ in ${\displaystyle {\mathfrak {g}}}$ induces a vector field ${\displaystyle \rho (\xi )}$ on ${\displaystyle M}$ describing the infinitesimal action of ${\displaystyle \xi }$. To be precise, at a point ${\displaystyle x}$ in ${\displaystyle M}$ the vector ${\displaystyle \rho (\xi )_{x}}$ is

where ${\displaystyle \exp :{\mathfrak {g}}\to G}$ is the exponential map and ${\displaystyle \cdot }$ denotes the ${\displaystyle G}$-action on ${\displaystyle M}$. Let ${\displaystyle \iota _{\rho (\xi )}\omega \,}$ denote the contraction of this vector field with ${\displaystyle \omega }$. Because ${\displaystyle G}$ acts by symplectomorphisms, it follows that ${\displaystyle \iota _{\rho (\xi )}\omega \,}$ is closed (for all ${\displaystyle \xi }$ in ${\displaystyle {\mathfrak {g}}}$).

Suppose that ${\displaystyle \iota _{\rho (\xi )}\omega \,}$ is not just closed but also exact, so that ${\displaystyle \iota _{\rho (\xi )}\omega =\mathrm {d} H_{\xi }}$ for some function ${\displaystyle H_{\xi }:M\to \mathbb {R} }$. If this holds, then one may choose the ${\displaystyle H_{\xi }}$ to make the map ${\displaystyle \xi \mapsto H_{\xi }}$ linear. A momentum map for the ${\displaystyle G}$-action on ${\displaystyle (M,\omega )}$ is a map ${\displaystyle \mu :M\to {\mathfrak {g}}^{*}}$ such that

for all ${\displaystyle \xi }$ in ${\displaystyle {\mathfrak {g}}}$. Here ${\displaystyle \langle \mu ,\xi \rangle }$ is the function from ${\displaystyle M}$ to ${\displaystyle \mathbb {R} }$ defined by ${\displaystyle \langle \mu ,\xi \rangle (x)=\langle \mu (x),\xi \rangle }$. The momentum map is uniquely defined up to an additive constant of integration (on each connected component).

An ${\displaystyle G}$-action on a symplectic manifold ${\displaystyle (M,\omega )}$ is called Hamiltonian if it is symplectic and if there exists a momentum map.

A momentum map is often also required to be ${\displaystyle G}$-equivariant, where ${\displaystyle G}$ acts on ${\displaystyle {\mathfrak {g}}^{*}}$ via the coadjoint action, and sometimes this requirement is included in the definition of a Hamiltonian group action. If the group is compact or semisimple, then the constant of integration can always be chosen to make the momentum 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-cocycle on the group with values in ${\displaystyle {\mathfrak {g}}^{*}}$, as first described by Souriau (1970).