Action-angle coordinates

Intuition [Beginner]

Action-angle coordinates is canonical coordinates near compact invariant tori. It gives a geometric rule for motion, constraint, or size without choosing ordinary distances as the main object.

The first picture is phase space: position and momentum are paired. A symplectic structure records how those pairs rotate into motion. It is less like a ruler and more like a turning rule.

This idea matters because Hamiltonian mechanics, reduction, and Floer theory all use the same pairing language.

Visual [Beginner]

The diagram shows a surface with arrows and level curves. It is a mnemonic for the way symplectic geometry ties motion to paired directions.

The picture is not a coordinate proof. It marks the objects that the formal definition makes precise.

Worked example [Beginner]

Use the plane with coordinates called position and momentum. A point records both where something is and how strongly it is moving.

For the energy rule "half position squared plus half momentum squared," the level curves are circles. The motion follows those circles instead of moving straight toward lower energy.

At the point with position 1 and momentum 0, the motion points in the momentum direction. After a quarter turn, the roles have exchanged.

What this tells us: symplectic geometry turns an energy rule into organized motion.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $M$ denote a smooth manifold or $V$ a finite-dimensional vector space, according to the context. The prerequisites used here are 05.02.03, 05.01.04. The concept action-angle coordinates is the structure described by the following data: a symplectic form $\omega$, compatible maps or subspaces, and the equations preserving the relevant pairing.

For this unit, the sign convention is the geometric Hamiltonian convention

$$\omega (X_H,-)=dH.$$

When the unit is purely linear, replace differential forms by skew bilinear forms. When a group acts, infinitesimal generators are paired with Hamiltonian functions through the same equation ^{[Arnold §50]}.

A morphism between such structures is a smooth or linear map preserving the specified symplectic data. This preservation condition is the source of rigidity results absent from ordinary volume geometry.

Key theorem with proof [Intermediate+]

Theorem (Liouville-Arnold). Let $(M^{2n},\omega )$ be a symplectic manifold and let $F=(F_1,\dots ,F_n):M\to {\mathbb{R}}^n$ be smooth functions with $\{F_i,F_j\}=0$ for all $i,j$. Fix a regular value $c\in {\mathbb{R}}^n$ of $F$ — i.e., $d{F}_1,\dots ,d{F}_n$ are linearly independent at every point of $M_c:=F^{-1}(c)$ — and assume $M_c$ is compact and connected. Then:

1. $M_c$ is diffeomorphic to the $n$-torus $T^n={\mathbb{R}}^n/{\mathbb{Z}}^n$.
2. On a neighbourhood $U\supset M_c$ in $M$ there exist coordinates $(I_1,\dots ,I_n,{\theta }^1,\dots ,{\theta }^n)$, with ${\theta }^i\in \mathbb{R}/2\pi \mathbb{Z}$ and $I_i$ a smooth function of $c$ alone, such that $$\omega {|}_U=\sum _{i=1}^{n}d{I}_i\wedge d{\theta }^i.$$
3. In these coordinates, the Hamiltonian flow of any function $H=H(I)$ is linear: ${\dot{I}}_i=0$, ${\dot{\theta }}^i=\partial H/\partial I_i$ ^{[Arnold §49; ref: TODO_REF Cannas da Silva Lecture 24]}.

Proof.

Step 1: $M_c$ is Lagrangian. The tangent space at $p\in M_c$ is $T_p{M}_c={\bigcap }_i\ker d{F}_i(p)$, of dimension $n$. The Hamiltonian vector fields $X_{F_i}$ satisfy ${\iota }_{X_{F_i}}\omega =d{F}_i$, so $X_{F_i}\in (T_p{M}_c{)}^{\omega }$ — the $\omega$-orthogonal complement, also of dimension $n$ by nondegeneracy. Poisson-commutativity reads $\omega (X_{F_i},X_{F_j})=\{F_j,F_i\}=0$, and $X_{F_i}(F_j)=\{F_j,F_i\}=0$ shows $X_{F_i}\in T_p{M}_c$. The $X_{F_i}$ are linearly independent at each $p\in M_c$ because the $d{F}_i$ are, hence span $T_p{M}_c$. So $T_p{M}_c\subseteq (T_p{M}_c{)}^{\omega }$, and dimension count gives $T_p{M}_c=(T_p{M}_c{)}^{\omega }$: $M_c$ is Lagrangian and $\omega {|}_{M_c}=0$.

Step 2: a free abelian action on $M_c$. The flows ${\varphi }_t^i$ of the $X_{F_i}$ commute because $[X_{F_i},X_{F_j}]=X_{\{F_i,F_j\}}=0$, and each preserves every $F_j$ (since $X_{F_i}(F_j)=0$), hence preserves $M_c$. Compactness of $M_c$ ensures completeness of these flows on $M_c$. Define $$\Psi :{\mathbb{R}}^n\times M_c\to M_c,\Psi (t,p)=({\varphi }_{t_1}^1\circ \cdots \circ {\varphi }_{t_n}^n)(p).$$ This is a smooth ${\mathbb{R}}^n$-action. Its infinitesimal generators are the $X_{F_i}$, which span $T{M}_c$ at every point, so the orbit map ${\mathbb{R}}^n\to M_c$, $t\mapsto \Psi (t,p)$, is a local diffeomorphism near $0$. The action is locally free.

Step 3: stabilisers are full-rank lattices. Fix $p\in M_c$ and set ${\Gamma }_p:=\{t\in {\mathbb{R}}^n:\Psi (t,p)=p\}$. Local freeness makes ${\Gamma }_p\subset {\mathbb{R}}^n$ discrete; smoothness makes it a closed subgroup. The orbit ${\mathbb{R}}^n\cdot p\cong {\mathbb{R}}^n/{\Gamma }_p$ is open in $M_c$ (local diffeomorphism) and closed (image of compact set under a continuous map into a Hausdorff space — apply to closures of bounded fundamental domains; equivalently, the orbit of a compact ${\mathbb{R}}^n$-action is closed). Connectedness of $M_c$ forces $M_c={\mathbb{R}}^n/{\Gamma }_p$. Compactness of $M_c$ then forces ${\Gamma }_p$ to have rank exactly $n$: a lattice of lower rank produces a non-compact quotient. So ${\Gamma }_p\cong {\mathbb{Z}}^n$ and $M_c\cong T^n$, proving (1).

Step 4: action coordinates from period integrals. Since $\omega$ is closed, on a tubular neighbourhood $U\supset M_c$ we may write $\omega =d\lambda$ for a primitive $\lambda$ (e.g., from the Weinstein tubular neighbourhood of a Lagrangian $T^n$, $U\simeq T^{\ast }{T}^n$ near the zero section, with $\lambda$ the canonical Liouville form). Pick a smoothly varying basis ${\gamma }_1(c),\dots ,{\gamma }_n(c)$ of $H_1(M_c;\mathbb{Z})$ (extend a basis chosen at one regular value across nearby regular values by parallel transport along the foliation). Set $$I_i(c):=\frac{1}{2\pi }{\oint }_{{\gamma }_i(c)}\lambda .$$ Independence of the choice of primitive: replacing $\lambda$ by $\lambda +d\sigma$ adds ${\oint }_{{\gamma }_i}d\sigma =0$ (closed loop). Independence of homology representative: if ${\gamma }_i^{\prime }$ is homologous to ${\gamma }_i$ in $M_c$, the difference bounds a $2$-chain $\Sigma \subset M_c$ and Stokes gives ${\oint }_{{\gamma }_i^{\prime }}\lambda -{\oint }_{{\gamma }_i}\lambda ={\int }_{\Sigma }\omega =0$ since $\omega {|}_{M_c}=0$. Smoothness of $c\mapsto I_i(c)$ follows from smooth dependence of ${\gamma }_i(c)$ and $\lambda$.

Step 5: the $I_i$ are independent and Poisson-commute with each $F_j$. The Jacobian $\partial I/\partial c$ is invertible (a calculation: differentiating ${\oint }_{{\gamma }_i}\lambda$ in $c$ yields $\partial I_i/\partial c_j={\tau }_{ij}/2\pi$ where ${\tau }_{ij}$ is the $j$-th time required to traverse ${\gamma }_i$ under the flow of $X_{F_j}$ — and these times form an invertible matrix because the $X_{F_j}$ are linearly independent and ${\gamma }_i$ is a homologically nonzero loop). So $(I_1,\dots ,I_n)$ is an alternative set of independent integrals, with $\{I_i,I_j\}=0$ inherited from $\{F_i,F_j\}=0$.

Step 6: angle coordinates from the action. Let $X_i:=X_{I_i}$. The flow times $\tau$ that close orbits along ${\gamma }_i$ satisfy $({\tau }_1,\dots ,{\tau }_n)=(2\pi ,0,\dots ,0)$ for the ${\gamma }_1$-direction and analogously for the others — this is the content of the action definition: $I_i$ is dual to ${\gamma }_i$. So the lattice $\Gamma$ in the $(I,X)$-frame is the standard lattice $2\pi {\mathbb{Z}}^n$. Choose any base point $p_0(c)\in M_c$ depending smoothly on $c$, and define ${\theta }^i(p)\in \mathbb{R}/2\pi \mathbb{Z}$ by: ${\theta }^i$ is the time of the $X_i$-flow needed to reach $p$ from $p_0$, modulo $2\pi$. Then $X_i=\partial /\partial {\theta }^i$.

Step 7: $(I,\theta )$ are Darboux. By construction $X_{I_i}=\partial /\partial {\theta }^i$, so ${\iota }_{\partial /\partial {\theta }^i}\omega =d{I}_i$. The angles span $T{M}_c$, on which $\omega$ vanishes, so $\omega (\partial /\partial {\theta }^i,\partial /\partial {\theta }^j)=0$. To show $\omega (\partial /\partial I_i,\partial /\partial I_j)=0$, observe that $\omega =d\lambda$ on $U$ and $\lambda =\sum I_{i}d{\theta }^i+d\sigma$ for some function $\sigma$ on $U$ (since $\sum I_{i}d{\theta }^i$ has the same period integrals as $\lambda$ along the ${\gamma }_i$, and any closed $1$-form on $T^n$ with vanishing periods is exact). Then $$\omega =d\lambda =\sum _{i}d{I}_i\wedge d{\theta }^i,$$ proving (2). Hamilton's equations for $H=H(I)$ become ${\dot{I}}_i=-\partial H/\partial {\theta }^i=0$ and ${\dot{\theta }}^i=\partial H/\partial I_i$, which are constants depending only on $I$. The flow on each torus $\{I=\text{const}\}$ is linear at frequency $\omega (I):=\partial H/\partial I$, proving (3). $\square$

Bridge. The construction here builds toward 05.08.01 (arnold conjecture and floer homology setup), where the same data is developed in the next layer of the strand. The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Advanced results [Master]

The construction of action-angle coordinates is invariant under symplectomorphism. In local Darboux coordinates, the form is modeled by

$${\omega }_0=\sum _{i}d{q}_i\wedge d{p}_i,$$

and global information is carried by the way these local models are glued. This separation between local normal form and global obstruction is a recurring feature of the subject ^{[Arnold §50]}.

For Hamiltonian group actions, the infinitesimal action, moment map, and Poisson bracket form one algebraic package. The identity $\omega (X_H,-)=dH$ converts functions into vector fields, and equivariance converts Lie brackets into Poisson brackets. Reduction, coadjoint orbits, and Floer complexes are built from this package.

Compactness and transversality questions enter when one counts trajectories or curves. In the finite-dimensional part of the strand, the essential inputs are closedness, nondegeneracy, and regular-value hypotheses. In Floer-theoretic units, analytic compactness replaces finite-dimensional regularity.

Synthesis. This construction generalises the pattern fixed in 05.02.03 (integrable system), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

Proposition. Hamiltonian vector fields preserve the symplectic form.

Let $X_H$ be defined by ${\iota }_{X_H}\omega =dH$. Cartan's formula gives

$${\mathcal{L}}_{X_H}\omega =d{\iota }_{X_H}\omega +{\iota }_{X_H}d\omega =d(dH)+0=0.$$

Hence the derivative of ${\Phi }_t^{\ast }\omega$ along the Hamiltonian flow is zero, and ${\Phi }_t^{\ast }\omega =\omega$ whenever the flow is defined.

Proposition. A symplectic linear map preserves symplectic orthogonals.

Let $A:V\to V$ satisfy $\omega (Av,Aw)=\omega (v,w)$. If $x\in W^{\omega }$, then $\omega (Ax,Aw)=\omega (x,w)=0$ for every $w\in W$. Hence $Ax\in (AW{)}^{\omega }$. Applying the same argument to $A^{-1}$ gives equality.

Connections [Master]

• The smooth-manifold language comes from 03.02.01, and differential forms enter through 03.04.02.

• The closedness condition uses exterior derivative 03.04.04 and feeds de Rham cohomology 03.04.06.

• This unit connects directly to 05.02.03, 05.01.04, and 05.08.01 inside the symplectic strand.

• Hamiltonian action principles also connect to variational calculus 03.04.08.

Historical & philosophical context [Master]

Hamiltonian mechanics supplied the original phase-space formalism, with canonical coordinates and the pairing of position and momentum. Poincare's qualitative theory of dynamical systems and Arnold's geometric mechanics placed this formalism in the language of manifolds and differential forms ^[Arnold].

Gromov's 1985 introduction of pseudoholomorphic curves changed symplectic topology by producing global rigidity phenomena not visible from Darboux's local theorem ^{[Gromov 1985]}. Floer's work later adapted infinite-dimensional Morse theory to Hamiltonian fixed points and Lagrangian intersections ^{[Floer original papers]}.

Bibliography [Master]

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