the moment map
Noether's principle says every continuous symmetry of a mechanical system gives a conserved quantity: rotational symmetry gives angular momentum, translational symmetry gives linear momentum. The moment map is the geometric machine that produces ALL these conserved quantities at once. When a symmetry group G acts on a symplectic manifold, the moment map is a single map from the manifold to the dual of the Lie algebra of G, whose components are exactly the conserved momenta generating each one-parameter subgroup of symmetries.
Precisely, suppose a Lie group G acts on (M, omega) by symplectomorphisms, with Lie algebra g. Each xi in g generates a vector field X_xi on M (the infinitesimal action). The action is Hamiltonian if there is a moment map mu: M -> g^* (the dual of the Lie algebra) such that for every xi in g, the function <mu, xi> (pairing mu with xi, a function on M) is a Hamiltonian for X_xi: that is, d<mu, xi> = iota_{X_xi} omega. One usually also requires mu to be equivariant for the coadjoint action of G on g^*. Concretely each component of mu is the conserved quantity attached to one symmetry direction. For SO(3) rotating R^6, mu is literally the angular momentum vector; for translations it is linear momentum — the names 'moment map' and 'momentum map' come from exactly these.
Why it matters: the moment map is the hinge of modern symmetry reduction. Its level set mu^{-1}(0), quotiented by G, is the reduced phase space of Marsden-Weinstein reduction, the rigorous way to 'divide out' a symmetry. It encodes Noether's theorem (mu is constant along the Hamiltonian flow whenever H is G-invariant, since the components Poisson-commute with H), and through convexity theorems (Atiyah, Guillemin-Sternberg) the image of mu for a torus action is a convex polytope, linking symplectic geometry to combinatorics and toric varieties. A caveat: not every symplectic action is Hamiltonian — a moment map need not exist (there can be a cohomological obstruction in H^1(M) or H^2(g)), and when it exists it may be unique only up to a constant (shift by a central element) and may fail to be equivariant without adjustment.
Let SO(3) act on the phase space R^6 = T*R^3 of a particle in space by simultaneously rotating positions q and momenta p. This action is Hamiltonian, and its moment map mu: R^6 -> so(3)^* = R^3 is mu(q, p) = q x p, the angular momentum vector. Each component <mu, xi> generates rotation about the corresponding axis, and if the Hamiltonian H is rotation-invariant then angular momentum q x p is conserved — Noether's theorem, read directly off the moment map.
For SO(3) on R^6, mu(q, p) = q x p is literally angular momentum — Noether made geometric.
A symplectic action need not admit a moment map: existence can be obstructed (the action being merely symplectic, not Hamiltonian), and even when mu exists it is determined only up to an additive constant in g^* unless you impose equivariance. 'Symplectic action' and 'Hamiltonian action' are not synonyms.