In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map[1]) 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.
be a manifold with symplectic form
Suppose that a Lie group
describing the infinitesimal action of
denote the contraction of this vector field with
The momentum map is uniquely defined up to an additive constant of integration (on each connected component).
is called Hamiltonian if it is symplectic and if there exists a momentum map.
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).
In the case of a Hamiltonian action of the circle
, and the momentum map is simply the Hamiltonian function that generates the circle action.
is the Euclidean group generated by rotations and translations.
is a six-dimensional group, the semidirect product of
be its cotangent bundle, with projection map
is Hamiltonian with momentum map
denotes the contraction of the vector field
The facts mentioned below may be used to generate more examples of momentum maps.
Suppose that the action of a Lie group
is Hamiltonian, as defined above, with equivariant momentum map
The quotient inherits a symplectic form from
; that is, there is a unique symplectic form on the quotient whose pullback to
Thus, the quotient is a symplectic manifold, called the Marsden–Weinstein quotient, after (Marsden & Weinstein 1974), symplectic quotient, or symplectic reduction of
More generally, if G does not act freely (but still properly), then (Sjamaar & Lerman 1991) showed that
is a stratified symplectic space, i.e. a stratified space with compatible symplectic structures on the strata.
on a surface carries an infinite dimensional symplectic form The gauge group
Then the map that sends a connection to its curvature is a moment map for the action of the gauge group on connections.
In particular the moduli space of flat connections modulo gauge equivalence