Action-angle coordinates

On an open set $U$ of a $2n$ -dimensional symplectic manifold, action-angle coordinates are a diffeomorphism $U\cong B\times\mathbb T^n$ , where $B\subset\mathbb R^n$ is open, providing variables

(I_1,\ldots,I_n,\theta_1,\ldots,\theta_n)\in B\times\mathbb T^n

such that $\omega=\sum_{j=1}^n dI_j\wedge d\theta_j$ and the functions of a specified completely integrable system depend only on $I=(I_1,\ldots,I_n)$ . The $I_j$ are action variables and the circle-valued $\theta_j$ are angle variables. In particular, if $H=H(I)$ , Hamilton's equations give $\dot I_j=0$ and $\dot\theta_j=\partial H/\partial I_j$ , so the motion on each torus $I=\mathrm{constant}$ is linear.

Geometric meaning

The projection $B\times\mathbb T^n\to B$ describes a local fibration by invariant Lagrangian tori. Actions may be obtained by integrating a local primitive of the symplectic form around a basis of one-cycles on each torus; changing that integral basis transforms the actions by an integral affine change. Angles then parametrize the commuting flows around the torus.

Existence and limitations

The Liouville–Arnold theorem supplies action-angle coordinates near a compact connected regular fiber of the integral map. It is a local statement around a torus, not a guarantee of one coordinate system over the entire regular locus. Monodromy of the period lattice and the topology of the torus fibration can obstruct global action-angle coordinates.

Examples and conventions

The one-dimensional harmonic oscillator admits an action proportional to its energy and an angle equal to the phase of its periodic orbit away from the equilibrium. The equilibrium itself is a singular fiber, so these angle coordinates do not extend through it. Some authors write $\omega=\sum_j d\theta_j\wedge dI_j$ ; the corresponding Hamilton-equation signs change with that convention.

References

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, 2nd ed., Springer, 1989. DOI record. Relevant: Chapter 10, action-angle variables and complete integrability.
J. J. Duistermaat, “On global action-angle coordinates,” Communications on Pure and Applied Mathematics 33 (1980), 687–706. DOI record. Relevant: §§1–3, local coordinates and global obstructions.