Hamiltonian Systems and Moment Maps

A Hamiltonian system is a symplectic manifold $(M, ω)$ together with a smooth function $H$ generating a flow that preserves $ω$ . A moment map $μ : M \to g^{*}$ packages the conserved quantities arising from a Lie group symmetry $G$ acting by symplectomorphisms, providing the geometric form of Noether's theorem. Used throughout classical mechanics, geometric quantization, integrable systems, and gauge theory. Foundational ideas due to Hamilton (1830s), Lie, Noether (1918), with the modern formulation by Souriau, Kostant, and Kirillov (1960s–70s).

A symplectic manifold $(M, ω)$ is an even-dimensional smooth manifold equipped with a closed, nondegenerate 2-form $ω$ . Nondegeneracy lets every function $H$ generate a unique vector field $X_{H}$ via $ι_{X_{H}} ω = d H$ , whose flow preserves both $ω$ and $H$ — this is Hamiltonian dynamics in coordinate-free form. When a Lie group $G$ acts on $M$ preserving $ω$ , each Lie algebra element induces a Hamiltonian vector field, and assembling their generating functions into a single $g^{*}$ -valued map yields the moment map $μ$ , an equivariant geometric encoding of all conserved momenta. The surprising fact: the moment map turns the messy bookkeeping of symmetry and conservation into a single intertwining map whose level sets and quotients (symplectic reduction) construct nearly every interesting phase space in physics and geometry.

The Core Insight

Symplectic structure converts functions into flows ( $H \mapsto X_{H}$ ) and symmetries into momenta ( $ξ \mapsto μ^{ξ}$ ). The moment map is the bridge: it is simultaneously the Noether conserved quantity and a $G$ -equivariant map whose fibers, after quotienting by the symmetry, carry their own induced symplectic structure. Conservation, symmetry, and dimensional reduction are one phenomenon.

Formal Statement

Symplectic Manifold and Hamiltonian Vector Field

A symplectic manifold is a pair $(M, ω)$ where $ω \in Ω^{2} (M)$ satisfies

d ω = 0, ω^{n} \neq = 0 (dim M = 2 n) .

⟳closedness is the integrability condition;

ω^{n} \neq = 0

is equivalent to nondegeneracy of

ω_{p}

as a bilinear form at each point.

Nondegeneracy gives a bundle isomorphism $ω^{♭} : T M \to T^{*} M$ , $v \mapsto ι_{v} ω$ . For $H \in C^{\infty} (M)$ , the Hamiltonian vector field $X_{H}$ is defined by

ι_{X_{H}} ω = d H .

⟳invert

ω^{♭}

applied to the 1-form

d H

; uniqueness follows from nondegeneracy.

Poisson Bracket

The Poisson bracket of $f, g \in C^{\infty} (M)$ is

{f, g} = ω (X_{f}, X_{g}) = X_{f} (g) = - X_{g} (f) .

⟳evaluate

ι_{X_{f}} ω

X_{g}

and use [2]; the sign conventions follow from antisymmetry of

ω

It is bilinear, antisymmetric, satisfies the Jacobi identity (equivalent to $d ω = 0$ ), and the Leibniz rule, making $(C^{\infty} (M), {\cdot, \cdot})$ a Poisson algebra.

Conservation of $\omega$ and $H$

The flow $ϕ_{t}$ of $X_{H}$ satisfies $ϕ_{t}^{*} ω = ω$ and $H \circ ϕ_{t} = H$ . Thus Hamiltonian flows are symplectomorphisms preserving energy.

Step 1. By Cartan's magic formula,

L_{X_{H}} ω = d ι_{X_{H}} ω + ι_{X_{H}} d ω

. Step 2. Use [2] to replace

ι_{X_{H}} ω = d H

and [1] to drop

d ω = 0

L_{X_{H}} ω = d (d H) + ι_{X_{H}} (0) = 0.

⟳

d^{2} = 0

kills the first term, closedness the second.

Step 3. Since the Lie derivative along

X_{H}

ω

vanishes,

\frac{d}{d t} ϕ_{t}^{*} ω = ϕ_{t}^{*} L_{X_{H}} ω = 0

, so

ϕ_{t}^{*} ω = ω

. Step 4. Energy conservation:

X_{H} (H) = {H, H} = 0

by antisymmetry of [3], so

H

is constant along trajectories.

■

Symplectic, Hamiltonian, and Equivariant Moment Map

Let a Lie group $G$ with Lie algebra $g$ act on $(M, ω)$ by symplectomorphisms. Each $ξ \in g$ generates a fundamental vector field $ξ_{M}$ (the infinitesimal generator). The action is Hamiltonian if there is a smooth map $μ : M \to g^{*}$ such that, writing $μ^{ξ} (p) := ⟨ μ (p), ξ ⟩$ ,

d μ^{ξ} = ι_{ξ_{M}} ω, \forall ξ \in g .

⟳this says

X_{μ^{ξ}} = ξ_{M}

, i.e.

μ^{ξ}

is the Hamiltonian generating the symmetry direction

ξ

μ

is equivariant when it intertwines the

G

-action and the coadjoint action

Ad^{*}

μ (g \cdot p) = Ad_{g^{- 1}}^{*} μ (p) .

⟳differentiate the desired naturality of generating functions under

g

; equivalently

{μ^{ξ}, μ^{η}} = μ^{[ξ, η]}

infinitesimally.

Infinitesimal equivariance is the comoment condition

{μ^{ξ}, μ^{η}} = μ^{[ξ, η]} .

⟳apply [5] twice and use

[ξ_{M}, η_{M}] = - [ξ, η]_{M}

plus the Poisson-bracket relation

{μ^{ξ}, μ^{η}} = ω (ξ_{M}, η_{M})

Noether: Moment Map Components are Conserved

If $H$ is $G$ -invariant ( $H \circ g = H$ for all $g \in G$ ), then $μ$ is conserved by the Hamiltonian flow of $H$ : each $μ^{ξ}$ is a constant of motion, ${H, μ^{ξ}} = 0$ .

Step 1.

G

-invariance of

H

means

ξ_{M} (H) = 0

for all

ξ

, since

ξ_{M}

is the infinitesimal generator of the symmetry. Step 2. Compute

{H, μ^{ξ}} = X_{H} (μ^{ξ}) = - X_{μ^{ξ}} (H)

by antisymmetry of [3]. Step 3. By [5],

X_{μ^{ξ}} = ξ_{M}

, hence

{H, μ^{ξ}} = - ξ_{M} (H) = 0.

⟳substitute

X_{μ^{ξ}} = ξ_{M}

from [5] and apply Step 1.

Therefore $\frac{d}{d t} μ^{ξ} = X_{H} (μ^{ξ}) = 0$ along trajectories. $■$

Marsden–Weinstein Symplectic Reduction

Let $μ$ be an equivariant moment map and $η \in g^{*}$ a regular value with $G_{η}$ (the coadjoint stabilizer) acting freely and properly on $μ^{- 1} (η)$ . Then the quotient $M_{η} := μ^{- 1} (η) / G_{η}$ is a smooth symplectic manifold with a unique $ω_{η}$ satisfying

π^{*} ω_{η} = ι^{*} ω,

where

ι : μ^{- 1} (η) ↪ M

and

π : μ^{- 1} (η) \to M_{η}

. Moreover

dim M_{η} = dim M - dim G - dim G_{η}

. Step 1. Regularity makes

μ^{- 1} (η)

a smooth submanifold of codimension

dim G

. Its tangent space is

ker d μ_{p} = (g \cdot p)^{ω}

, the symplectic orthogonal of the orbit directions (using [5]:

d μ^{ξ} = ι_{ξ_{M}} ω

means

v \in ker d μ ⟺ ω (ξ_{M}, v) = 0 \forall ξ

). Step 2. The tangent to the

G_{η}

-orbit inside

μ^{- 1} (η)

is the characteristic distribution

g_{η} \cdot p = T_{p} μ^{- 1} (η) \cap (null space of ι^{*} ω)

. The freely/properly acting

G_{η}

makes

π

a principal bundle. Step 3. Define

ω_{η}

T_{[p]} M_{η} ≅ T_{p} μ^{- 1} (η) / g_{η} \cdot p

ω_{η} ([u], [v]) := ω (u, v)

. Well-definedness: the radical of

ι^{*} ω

is exactly

g_{η} \cdot p

, so the value is independent of representatives. Step 4.

ω_{η}

is nondegenerate (we quotiented by the radical) and closed:

d (ι^{*} ω) = ι^{*} d ω = 0 = π^{*} d ω_{η}

and

π^{*}

is injective on forms. This yields [9].

■

Emmy Noether | 1882–1935

Noether's 1918 theorems connected continuous symmetries to conservation laws in field theory and mechanics, the analytic ancestor of the moment map. Working at Göttingen (initially unpaid, lecturing under Hilbert's name due to discrimination against women), she also revolutionized abstract algebra with her theory of ideals and modules. The moment map is the modern, coordinate-free, group-theoretic crystallization of "Noether's first theorem": each independent symmetry generator contributes one component $μ^{ξ}$ , a conserved current integrated over phase space.

Building Intuition

Think of $ω$ as a currency converter between velocities (vectors) and forces (covectors). The relation $ι_{X_{H}} ω = d H$ says: the gradient-like covector $d H$ is exchanged for the velocity $X_{H}$ that flows along level sets of $H$ (not across them) — hence energy is conserved automatically. For symmetry: a one-parameter group of symplectomorphisms is itself a Hamiltonian flow, and its generating function is the conserved momentum. Stack all these momenta for every symmetry direction $ξ$ into one map valued in $g^{*}$ , and you get $μ$ . Equivariance [6] is the statement that "rotating the system rotates its angular momentum vector the same way" — the bookkeeping is consistent. Reduction is the geometric act of fixing the conserved momentum and forgetting the symmetry directions; what remains is a smaller phase space with its own dynamics — like passing to the center-of-mass frame.

Moment map level set and symplectic reduction

<svg viewBox="0 0 640 380" xmlns="http://www.w3.org/2000/svg">
<rect width="640" height="380" fill="#0a0e17"/>
<text x="20" y="30" fill="#e2e8f0" font-size="16">Symplectic reduction: fix μ, quotient by G_η</text>
<!-- big manifold M -->
<ellipse cx="200" cy="210" rx="160" ry="120" fill="none" stroke="#94a3b8" stroke-width="2"/>
<text x="60" y="120" fill="#e2e8f0" font-size="14">(M, ω)</text>
<!-- level set mu^-1(eta) -->
<path d="M 100 210 C 140 130, 260 130, 300 210 C 260 290, 140 290, 100 210 Z" fill="none" stroke="#d4a853" stroke-width="2.5"/>
<text x="150" y="200" fill="#d4a853" font-size="13">μ⁻¹(η)</text>
<!-- orbit directions -->
<line x1="175" y1="160" x2="225" y2="160" stroke="#94a3b8" stroke-width="1" marker-end="url(#a)"/>
<line x1="175" y1="260" x2="225" y2="260" stroke="#94a3b8" stroke-width="1" marker-end="url(#a)"/>
<text x="230" y="164" fill="#e2e8f0" font-size="11">G_η-orbit</text>
<!-- arrow pi -->
<line x1="380" y1="210" x2="460" y2="210" stroke="#94a3b8" stroke-width="2" marker-end="url(#a)"/>
<text x="400" y="200" fill="#e2e8f0" font-size="13">π</text>
<!-- reduced space -->
<ellipse cx="540" cy="210" rx="70" ry="50" fill="none" stroke="#d4a853" stroke-width="2.5"/>
<text x="490" y="290" fill="#e2e8f0" font-size="14">M_η = μ⁻¹(η)/G_η</text>
<text x="500" y="215" fill="#d4a853" font-size="12">(reduced)</text>
<!-- mu map arrow to g* -->
<line x1="200" y1="335" x2="200" y2="365" stroke="#94a3b8" stroke-width="1.5" marker-end="url(#a)"/>
<text x="210" y="360" fill="#e2e8f0" font-size="12">μ : M → 𝔤*</text>
<defs>
<marker id="a" markerWidth="8" markerHeight="8" refX="6" refY="3" orient="auto">
<path d="M0,0 L6,3 L0,6 Z" fill="#94a3b8"/>
</marker>
</defs>
</svg>

Worked Examples

Angular momentum as the $SO(3)$ moment map

Given:

M = T^{*} R^{3} = R^{6}

with coordinates

(q, p)

, canonical

ω = \sum_{i} d p_{i} \land d q_{i}

. Let

G = S O (3)

act by rotating

q

(lifted to

p

). Step 1. A generator

ξ \in so (3) ≅ R^{3}

acts as

ξ_{M}

with

q \mapsto ξ \times q

p \mapsto ξ \times p

. Step 2. Seek

μ^{ξ}

with

X_{μ^{ξ}} = ξ_{M}

. Guess the angular momentum

μ (q, p) = q \times p \in R^{3} ≅ so (3)^{*}

, so

μ^{ξ} = (q \times p) \cdot ξ

. Step 3. Verify [5]:

d μ^{ξ} = (d q \times p + q \times d p) \cdot ξ

. Pairing

ι_{ξ_{M}} ω

with a test vector reproduces this; one checks component-wise, e.g. for

ξ = e_{3}

rotation about

z

μ^{e_{3}} = q_{1} p_{2} - q_{2} p_{1} = L_{z}

, and

ι_{ξ_{M}} ω = d L_{z}

. ✓ Step 4. Equivariance [6]:

μ (R q, R p) = (R q) \times (R p) = R (q \times p)

for

R \in S O (3)

, matching

Ad^{*} = R

R^{3}

. ✓ Boxed answer:

μ (q, p) = L = q \times p [units: kg\cdotm^{2} / s]

H = \frac{∣ p ∣ ^{2}}{2 m} + V (∣ q ∣)

is rotationally invariant, all three components of

L

are conserved.

$U(1)$ moment map and reduction of $\mathbb{C}^2$ to $\mathbb{CP}^1$

Given:

M = C^{2}

with

ω = \frac{i}{2} \sum_{k = 1}^{2} d z_{k} \land d \overset{z}{ˉ}_{k} = \sum_{k} d x_{k} \land d y_{k}

, and

U (1)

acting by

z \mapsto e^{i θ} z

. Step 1. Generator

ξ = 1 \in u (1) ≅ R

;

ξ_{M}

rotates each

z_{k}

, i.e. the vector field

\sum_{k} (x_{k} \partial_{y_{k}} - y_{k} \partial_{x_{k}})

. Step 2. Solving

d μ = ι_{ξ_{M}} ω

gives

μ (z) = \frac{1}{2} ∣ z ∣^{2} = \frac{1}{2} (∣ z_{1} ∣^{2} + ∣ z_{2} ∣^{2}) \in R ≅ u (1)^{*} .

Step 3. Choose regular value

η = \frac{1}{2}

. Then

μ^{- 1} (\frac{1}{2}) = {∣ z ∣^{2} = 1} = S^{3}

, the unit 3-sphere. Step 4. Quotient by

U (1)

(free action on

S^{3}

M_{1/2} = S^{3} / U (1) = CP^{1} = S^{2}, dim = 4 - 1 - 1 = 2.

The induced

ω_{η}

is (a multiple of) the Fubini–Study form; reduction realizes the Hopf fibration

S^{1} ↪ S^{3} \to S^{2}

symplectically. This is the prototype of constructing

CP^{n}

as a symplectic quotient.

Common Misconceptions

A moment map need not exist, and need not be unique even when it does.

Existence: A symplectic $G$ -action gives only a closed 1-form $ι_{ξ_{M}} ω$ ; for $μ^{ξ}$ to exist this must be exact. Obstruction lies in $H^{1} (M; R)$ (per generator) and a Lie-algebra cohomology class in $H^{2} (g; R)$ for equivariance.
Uniqueness: If $μ$ works, so does $μ + c$ for any $c \in g^{*}$ that is $Ad^{*}$ -invariant (annihilating $[g, g]$ ). For semisimple $g$ , $H^{1} (g) = H^{2} (g) = 0$ , so existence is automatic and $μ$ is unique.
Do not confuse the momentum map $μ : M \to g^{*}$ with the momentum coordinate $p$ ; the former is global and group-theoretic.

Torus action on the 2-torus with no moment map

Take $M = T^{2} = R^{2} / Z^{2}$ with coordinates $(θ_{1}, θ_{2})$ and $ω = d θ_{1} \land d θ_{2}$ (an area form, symplectic). Let $G = S^{1}$ act by translation $θ_{1} \mapsto θ_{1} + t$ . Then $ξ_{M} = \partial_{θ_{1}}$ and $ι_{ξ_{M}} ω = d θ_{2}$ , which is closed but not exact on $T^{2}$ (it generates $H^{1} (T^{2})$ ). Hence no global function $μ$ with $d μ = d θ_{2}$ exists: the action is symplectic but not Hamiltonian. This shows the cohomological obstruction is real, not pedantic.

Connections

Moment maps sit at the crossroads of geometry and physics. The Poisson-algebra structure [3] is the classical shadow of operator commutators in Geometric Quantization, where $μ$ becomes the quantized symmetry generators. Symplectic reduction [9] generalizes to Poisson Geometry (Poisson reduction) and underlies Symplectic Toric Manifolds, where the moment map image is a Delzant polytope. The convexity of $μ (M)$ for torus actions (Atiyah–Guillemin–Sternberg) links to Morse Theory via the components $μ^{ξ}$ being perfect Morse–Bott functions. Coadjoint orbits with the Kirillov–Kostant–Souriau form are the elementary "particles" of Coadjoint Orbits and Representation Theory, and reduction relates to Geometric Invariant Theory through the Kempf–Ness theorem identifying symplectic and GIT quotients.

Real-World Applications

Celestial mechanics & the $n$ -body problem: $S E (3)$ symmetry yields the moment map giving conservation of total linear and angular momentum; reduction passes to the center-of-mass and "reduced" two-body Kepler problem.
Rigid body dynamics: The Euler equations live on $so (3)^{*}$ as the reduction of $T^{*} S O (3)$ by left $S O (3)$ ; coadjoint orbits (angular-momentum spheres) are the reduced phase spaces.
Plasma physics & fluid dynamics: The Euler equations of an ideal fluid are Lie–Poisson dynamics on $g^{*}$ for $G =$ volume-preserving diffeomorphisms (Arnold); vorticity is the moment map.
Gauge theory: The Yang–Mills moment map sends a connection to the curvature; flat connections are $μ^{- 1} (0)$ , and their moduli space is a symplectic quotient (Atiyah–Bott).
Quantum spin & coherent states: $S U (2)$ coadjoint orbits ( $S^{2}$ ) are quantized to spin- $j$ representations via Borel–Weil; the moment map is the spin vector.
Image registration & computational anatomy: Diffeomorphism-group moment maps drive the EPDiff/LDDMM equations matching shapes.
Control theory & robotics: Reduction by symmetry simplifies optimal control on Lie groups (e.g., satellite attitude control using conserved angular momentum).

Historical Development

1834–35: W. R. Hamilton reformulates mechanics via the principle of least action and canonical equations, introducing the function $H$ and conjugate momenta.
1858–90: Sophus Lie develops the theory of continuous transformation groups and contact/symplectic transformations, with the bracket bearing his name.
1918: Emmy Noether proves her two theorems relating symmetries to conservation laws — the analytic seed of the moment map.
1927: H. Weyl and others formalize the algebra of canonical commutation relations, foreshadowing quantization of $μ$ .
1950s: A. Lichnerowicz and others advance global symplectic geometry on manifolds.
1962–70: B. Kostant and A. Kirillov independently introduce the orbit method and the canonical symplectic form on coadjoint orbits.
1966–70: J.-M. Souriau coins moment map (application moment) in Structure des systèmes dynamiques, giving the modern equivariant definition and the cohomological obstruction.
1974: J. Marsden and A. Weinstein publish Reduction of symplectic manifolds with symmetry, establishing Theorem [9].
1982: Atiyah and (independently) Guillemin–Sternberg prove the convexity theorem: $μ (M)$ for a torus action is the convex hull of the images of fixed points; Atiyah–Bott apply moment maps to gauge theory; Delzant (1988) classifies toric manifolds by their polytopes.

Subtleties & Limits

Several frontiers remain active:

Singular reduction: When $η$ is not a regular value or the action is not free, $M_{η}$ is a stratified symplectic space (Sjamaar–Lerman). A fully functorial, computationally tractable theory of singular quotients and their quantization is incomplete.
Quantization commutes with reduction ("[Q,R]=0"): Guillemin–Sternberg conjectured and (in compact cases) it is proven that quantizing $M$ then taking $G$ -invariants equals quantizing $M_{0}$ . Extensions to noncompact $M$ , singular $μ^{- 1} (0)$ , and infinite-dimensional groups are open.
Convexity beyond tori: Nonabelian convexity (Kirwan) describes $μ (M) \cap t_{+}^{*}$ ; sharp combinatorial control for general $G$ and the structure of multiplicity-free spaces is ongoing.
Moment maps in generalized geometry: Analogues for quasi-Hamiltonian, hyperkähler, and generalized complex structures, and Hitchin's "moment map" picture for special geometries, are not yet unified.
Infinite dimensions: Existence/equivariance obstructions and reduction for diffeomorphism and gauge groups require functional-analytic care that resists the clean finite-dimensional theorems.

Hamiltonian Systems and Moment Maps

The Core Insight

Formal Statement

Symplectic Manifold and Hamiltonian Vector Field

Poisson Bracket

Conservation of $\omega$ and $H$

Symplectic, Hamiltonian, and Equivariant Moment Map

Noether: Moment Map Components are Conserved

Marsden–Weinstein Symplectic Reduction

Emmy Noether | 1882–1935

Building Intuition

Moment map level set and symplectic reduction

Worked Examples

Angular momentum as the $SO(3)$ moment map

$U(1)$ moment map and reduction of $\mathbb{C}^2$ to $\mathbb{CP}^1$

Common Misconceptions

Torus action on the 2-torus with no moment map

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also

Before this

Where this leads