In mathematics, specifically in symplectic geometry, the momentum map is a tool associated with a Hamiltonianaction 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 symplecticquotients, discussed below, and symplectic cuts and sums.
Formal definition
Let M be a manifold with symplectic form ω. Suppose that a Lie group G acts on M via symplectomorphisms. Let be the Lie algebra of G, its dual, and the pairing between the two. Any ξ in induces a vector field ρ on M describing the infinitesimal action of ξ. To be precise, at a point x in M the vector is where is the exponential map and denotes the G-action on M. Let denote the contraction of this vector field with ω. Because G acts by symplectomorphisms, it follows that is closed. Suppose that is not just closed but also exact, so that for some function. Suppose also that the map sending is a Lie algebra homomorphism. Then a momentum map for the G-action on is a map such that for all ξ in. Here is the function from M to R defined by. The momentum map is uniquely defined up to an additive constant of integration. A momentum map is often also required to be G-equivariant, where G acts on via the coadjoint 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. The modification is by a 1-cocycle on the group with values in, as first described by Souriau.
The definition of the momentum map requires to be closed. In practice it is useful to make an even stronger assumption. The G-action is said to be Hamiltonianif and only if the following conditions hold. First, for every ξ in the one-form is exact, meaning that it equals for some smooth function If this holds, then one may choose the to make the map linear. The second requirement for the G-action to be Hamiltonian is that the map be a Lie algebra homomorphism from to the algebra of smooth functions on M under the Poisson bracket. If the action of G on is Hamiltonian in this sense, then a momentum map is a map such that writing defines a Lie algebra homomorphism satisfying. Here is the vector field of the Hamiltonian, defined by
Examples of momentum maps
In the case of a Hamiltonian action of the circle, the Lie algebra dual is naturally identified with, and the momentum map is simply the Hamiltonian function that generates the circle action. Another classical case occurs when is the cotangent bundle of and is the Euclidean group generated by rotations and translations. That is, is a six-dimensional group, the semidirect product of and. The six components of the momentum map are then the three angular momenta and the three linear momenta. Let be a smooth manifold and let be its cotangent bundle, with projection map. Let denote the tautological 1-form on. Suppose acts on. The induced action of on the symplectic manifold, given by for is Hamiltonian with momentum map for all. Here denotes the contraction of the vector field, the infinitesimal action of, with the 1-form. The facts mentioned below may be used to generate more examples of momentum maps.
Some facts about momentum maps
Let be Lie groups with Lie algebras, respectively. 1. Let be a coadjoint orbit. Then there exists a unique symplectic structure on such that inclusion map is a momentum map. 2. Let act on a symplectic manifold with a momentum map for the action, and be a Lie group homomorphism, inducing an action of on. Then the action of on is also Hamiltonian, with momentum map given by, where is the dual map to . A case of special interest is when is a Lie subgroup of and is the inclusion map. 3. Let be a Hamiltonian -manifold and a Hamiltonian -manifold. Then the natural action of on is Hamiltonian, with momentum map the direct sum of the two momentum maps and. Here, where denotes the projection map. 4. Let be a Hamiltonian -manifold, and a submanifold of invariant under such that the restriction of the symplectic form on to is non-degenerate. This imparts a symplectic structure to in a natural way. Then the action of on is also Hamiltonian, with momentum map the composition of the inclusion map with 's momentum map.
Symplectic quotients
Suppose that the action of a compact Lie groupG on the symplectic manifold is Hamiltonian, as defined above, with momentum map. From the Hamiltonian condition it follows that is invariant under G. Assume now that 0 is a regular value of μ and that G acts freely and properly on. Thus and its quotient are both manifolds. The quotient inherits a symplectic form from M; that is, there is a unique symplectic form on the quotient whose pullback to equals the restriction of ω to. Thus the quotient is a symplectic manifold, called the Marsden–Weinstein quotient, symplectic quotient or symplectic reduction of M by G and is denoted. Its dimension equals the dimension of M minus twice the dimension of G.
