Moment Maps, Symmetry & Marsden-Weinstein Reduction

From a single conserved quantity to a whole symmetry group

Guide 2 handed you the engine of Hamiltonian mechanics: on a symplectic manifold (M, omega), a smooth function H produces its Hamiltonian vector field X_H via iota_{X_H} omega = dH, and the Poisson bracket {f, H} measures how f changes along the flow of H. The cleanest fact there was the conservation principle: f is conserved along the flow of H exactly when {f, H} = 0, and by antisymmetry H is always conserved by its own flow. This is Noether's theorem in its leanest form — a symmetry generated by f that preserves the dynamics gives a conserved quantity f, and vice versa. The moment map is what you get when you run this not for one conserved quantity but for an entire connected group of symmetries at once.

Picture the simplest example, free rotation of a particle in the plane. The phase space is M = R^2 x R^2 with coordinates position q = (q_1, q_2) and momentum p = (p_1, p_2), carrying omega = dp_1 ^ dq_1 + dp_2 ^ dq_2. The circle group SO(2) acts by rotating q and p together by the same angle. The single number that this rotational symmetry conserves is the angular momentum mu = q_1 p_2 - q_2 p_1, and indeed {mu, H} = 0 for any rotationally invariant H. Here the symmetry group is one-dimensional, so one conserved number suffices. The whole content of the moment map is the upgrade from this scalar to a vector-valued quantity when the group is bigger.

What a moment map is, and the equation that defines it

Now let a Lie group G act on (M, omega) preserving omega — a symplectic action, the symmetry analogue of an isometry from the Riemannian rungs. Each element xi of the Lie algebra g (an 'infinitesimal symmetry', a direction in G near the identity) generates a vector field on M, written xi_M, by differentiating the action: xi_M at a point is the velocity of moving along the one-parameter subgroup exp(t xi). Because the action preserves omega, each xi_M is a symplectic vector field. A moment map is a smooth map mu: M -> g* (into the dual of the Lie algebra) that promotes every such xi_M to a genuine Hamiltonian vector field, with the conserved quantity for xi being the pairing of mu with xi.

Setup:   G acts on (M, omega),  omega-preserving
         xi in g  =>  vector field  xi_M  on M  (infinitesimal action)

Moment map  mu : M -> g*    defined by, for every xi in g,

   d< mu , xi >  =  iota_{xi_M} omega        ( <mu,xi> : M -> R  is the Hamiltonian for xi_M )

Equivariance:   mu( g . x )  =  Ad*_g ( mu(x) )    ( mu intertwines the G-action and coadjoint action )

Example (SO(2) rotating the plane):   g* = R,   mu = q_1 p_2 - q_2 p_1  =  angular momentum

The defining equation says each component <mu, xi> is the Hamiltonian generating the symmetry xi; equivariance ties mu to the coadjoint action Ad* on g*.

Read the defining equation slowly, because it is the whole idea in one line. For each xi, the real-valued function < mu, xi > on M is required to be a Hamiltonian whose vector field is xi_M, i.e. d< mu, xi > = iota_{xi_M} omega. So mu is a bookkeeping device: it stores, all at once, the entire family of conserved Hamiltonians, one for every direction in the symmetry algebra. Feeding mu a direction xi and a point x returns the value of that conserved quantity. The reason the target is g* and not g is exactly this pairing role — mu(x) is a linear functional eating Lie-algebra directions and returning the conserved number for each. Conservation is now automatic: if H is G-invariant, then each < mu, xi > Poisson-commutes with H, so every component of mu is a conserved quantity of the dynamics. That is Noether, vectorised.

Equivariance, coadjoint orbits, and when a moment map exists

The defining equation only pins down d< mu, xi >, the differential, so it fixes mu only up to a constant in g* for each xi — there is a genuine ambiguity. We tame it by demanding the natural compatibility with the group: mu should be equivariant, meaning it intertwines the G-action on M with the coadjoint action Ad* of G on g*, so mu(g . x) = Ad*_g(mu(x)). For SO(3) this just says the angular momentum vector rotates the way you rotate space, which is obviously true. Equivariance is the honest, geometric way to remove the constant ambiguity, and it is the hypothesis the reduction theorem will need. When G is compact or semisimple one can always arrange an equivariant moment map; in general there is a cohomological obstruction (a class in H^2 of the Lie algebra) to existence and to equivariance.

Equivariance forces a beautiful structure on the target. Because mu(g . x) = Ad*_g(mu(x)), the image of any G-orbit in M lands inside a single orbit of the coadjoint action in g*. These coadjoint orbits are themselves symplectic manifolds — the Kirillov-Kostant-Souriau form makes every orbit of Ad* in g* into a homogeneous symplectic G-space, and these orbits are the basic models out of which all 'simplest' Hamiltonian G-spaces are built. For G = SO(3) the coadjoint orbits in g* = R^3 are the spheres of fixed radius (plus the origin); the radius is the magnitude of angular momentum, and the sphere is the symplectic manifold S^2 you already know from Guide 1. So the abstract target g* is foliated by symplectic spheres, each one a phase space in its own right.

Be honest about hypotheses and pitfalls, because they are easy to drop. (1) A moment map need not exist: the action must be Hamiltonian, not merely symplectic. Each xi_M is symplectic (its contraction with omega is closed), but to be Hamiltonian that 1-form must be EXACT — the gap is measured by H^1(M), so on a manifold with nontrivial first cohomology the moment map can fail to exist (rotating a torus with its flat omega is the standard counterexample). (2) Even when it exists, equivariance can fail and may need a correction or a cocycle (the affine / non-equivariant case of Souriau). (3) Sign and factor conventions differ wildly across sources — whether omega = dp ^ dq or dq ^ dp, whether the defining equation is d<mu,xi> = iota_{xi_M} omega or its negative — so state your convention and expect other books to flip signs. None of this is proved here in detail; a careful treatment is a chapter of a mechanics or symplectic-geometry course.

Marsden-Weinstein reduction: quotienting out the symmetry

Here is the payoff. Symmetry should let you eliminate variables — angular momentum being conserved should collapse a three-dimensional problem to an effectively lower-dimensional one. Symplectic reduction, the Marsden-Weinstein-Meyer theorem of 1974, makes this precise and, crucially, keeps you inside the symplectic world. Start with a Hamiltonian G-action on (M, omega) with equivariant moment map mu. Pick a value, say 0 in g* (any coadjoint-fixed value works similarly). Then form the reduced space M_0 = mu^{-1}(0) / G: first restrict to the level set where the conserved quantities take the chosen value, then quotient by the leftover symmetry. The theorem says M_0 is again a symplectic manifold, with a canonical reduced form omega_0, and its dimension is dim M - 2 dim G.

Check 0 is a regular value of mu (use the regular value theorem: 0 is regular when G acts with no continuous stabilizer along mu^{-1}(0)). Then mu^{-1}(0) is a smooth submanifold of dimension dim M - dim G.
Observe the level set is coisotropic and its null directions are exactly the G-orbit directions: the kernel of omega restricted to mu^{-1}(0) is the tangent to the G-orbits. This is the geometric heart of the proof.
Quotient by G (assume the action is free and proper there, so the quotient is a smooth manifold). The degenerate directions of omega get collapsed precisely by the quotient, leaving a NONdegenerate form.
Define omega_0 on M_0 = mu^{-1}(0)/G by pulling back: pi*omega_0 = iota*omega, where iota is the inclusion of the level set and pi the quotient map. Check d omega_0 = 0 and nondegeneracy — done, (M_0, omega_0) is symplectic.

Why does the dimension drop by 2 dim G and not by dim G? This is the elegant part. The level set mu^{-1}(0) has dimension dim M - dim G — you used up dim G conditions setting the conserved quantities to zero. The level set turns out to be coisotropic, and the null directions of omega along it are exactly the directions of the G-orbits, another dim G. Quotienting by G removes precisely those degenerate directions, and what survives is nondegenerate. So you cut dim G going in (the constraint) and another dim G coming out (the quotient) — the two halves of a single symplectic-orthogonality fact. This is why physicists say each conserved quantity 'removes two degrees of freedom': the conserved value and its conjugate angle both disappear together.

Reduction in action, and honest limits

Make it concrete with the canonical example that builds projective space. Let G = U(1) act on M = C^{n+1} (real dimension 2n + 2, with its standard symplectic form) by the diagonal phase rotation z -> e^{i theta} z. The equivariant moment map is mu(z) = -(1/2)|z|^2 + const, essentially the squared length. Fix the level mu^{-1}(c) — a round sphere S^{2n+1} of fixed radius — and quotient by the circle action. The result is complex projective space CP^n, and the reduced symplectic form is (a multiple of) the Fubini-Study form. So CP^n, the basic symplectic (indeed Kahler) manifold from the complex-geometry rung, literally IS the Marsden-Weinstein reduction of flat C^{n+1} by the circle. Dimension count: (2n + 2) - 2(1) = 2n, exactly dim CP^n. This single construction is the prototype for symplectic and GIT quotients across geometry.

The general slogan is 'reduce at a value mu^{-1}(O)/G', and the value need not be 0. Reducing at a coadjoint orbit O instead of the single point 0 gives M_O = mu^{-1}(O)/G, still symplectic; this is how the SO(3) example at a fixed nonzero angular momentum produces the reduced phase space of a spinning system, and it is the rigorous version of 'fix the angular momentum and study the residual motion'. The whole framework also feeds Guide 5: contact reduction, symplectic cuts, and the symplectic topology of toric manifolds (where the moment map image is a convex polytope, the Atiyah-Guillemin-Sternberg theorem) all grow from this seed. Reduction is the bridge from the local rigidity of Guides 1 and 3 to global symplectic geometry.

Where reduction is delicate, say so plainly. (1) The clean theorem assumes 0 (or your chosen value) is a REGULAR value and that G acts FREELY and PROPERLY on the level set; drop freeness and the quotient develops orbifold or stratified singularities, handled by the harder Sjamaar-Lerman stratified reduction — not a quotient of smooth manifolds. (2) Compactness or properness of the action is what keeps the quotient Hausdorff and smooth; for noncompact G the quotient can be badly non-Hausdorff. (3) Reduction is genuinely a quotient, so it forgets information — you cannot in general reconstruct M from M_0 without the extra 'reconstruction' data (a connection), and reduced dynamics carries Berry-phase-like correction terms. (4) Like everything in this rung, we STATE and motivate Marsden-Weinstein; the full proof that omega_0 is well-defined, closed, and nondegenerate is a careful exercise in a graduate symplectic course, not something a guide proves.