Symplectic reduction and Marsden–Weinstein theory

Let G act symplectically on a 2n-manifold with an equivariant momentum map J. First use the value zero and assume it is regular, so Z=J^{-1}(0) is a submanifold of codimension dim G. Its tangent vectors annihilate every differential dJ_ξ. By the momentum identity, that means TZ is the symplectic orthogonal of the infinitesimal orbit directions. Equivariance at zero makes the orbit directions themselves tangent to Z. They therefore lie in the kernel of the restricted form, explaining why that restriction is presymplectic rather than symplectic.

Assume G acts freely and properly on Z. Freeness removes stabilizer directions, and properness ensures a well-behaved Hausdorff smooth orbit space. The quotient projection π:Z→Z/G is then a submersion. The restricted form is invariant and vanishes on orbit directions, so it descends to a unique two-form ω_red on the quotient. Its only null directions upstairs are the orbit directions being divided out; this proves nondegeneracy downstairs. Closedness descends from dω=0. These ingredients explain the theorem, although a full quotient-manifold construction is not reproduced here.

For an arbitrary regular value μ, equivariance says the full group can move one momentum level to another. Only the coadjoint stabilizer G_μ acts inside J^{-1}(μ). If this subgroup acts freely and properly on that level, the quotient J^{-1}(μ)/G_μ inherits the reduced symplectic form by the same pullback identity. Its dimension is 2n−dim G−dim G_μ. When μ is coadjoint fixed, including all values for an abelian group, G_μ=G and the familiar loss of two dimensions per symmetry returns.

If freeness fails but stabilizers are finite, a smooth-manifold conclusion generally has to be replaced by an orbifold statement. At singular momentum values the level itself can fail to be a manifold, and orbit types can produce strata of different dimensions. Even a visually simple quotient can hide these failures. A circle acting by rotations on a plane has a free action on every positive-radius circle but fixes the origin. Reducing its positive momentum levels gives points; the origin level does not satisfy the regular free hypotheses even though its set-theoretic quotient is also a point.

If H is G-invariant, its Hamiltonian flow preserves J and is equivariant, so it restricts to the momentum level and descends to the quotient. The reduced Hamiltonian is the function whose pullback equals H on that level. The pullback identity for ω verifies that the projected vector field satisfies the reduced Hamilton equations with the same sign convention. A conserved additive constant in H_red changes no motion, although it still affects reported energy values. For a noninvariant H, preservation and descent are not guaranteed by this theorem; a particular level can still admit reduced dynamics after those properties are checked directly.

Reduction removes symmetry-related descriptions while retaining their effect through the chosen momentum parameter. It does not remove every force associated with that parameter: in rotational mechanical examples a fixed angular momentum can create an effective potential. Reconstructing the original motion requires recovering the discarded group trajectory, not just lifting any reduced curve arbitrarily. This separation is useful in mechanics and gauge theories, but the finite-dimensional theorem stated here does not automatically settle functional-analytic issues of infinite-dimensional gauge quotients.