Action-angle coordinates - Imported Knowlpedia Content

Definition

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. Arnol'd, Chapter 10 gives the classical construction.

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 Duistermaat, §§1–3.

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.