---
tag: M3T81YNSKC
kind: concept
name: Action-angle coordinates
msc: [37J35, 70H06]
sources:
  - url: https://en.wikipedia.org/wiki/Action-angle_coordinates
same_as:
  wikidata: Q1000646
  wikipedia: Action-angle coordinates
---

## Summary

Action-angle coordinates are canonical coordinates $(I, \theta)$ for an [[integrable system|N0N7CMRCMH]] in which the Hamiltonian depends on the actions $I$ only.
In these coordinates the actions are constant and the angles turn at constant speed, so the equations of motion are solved at once.
They exist near compact regular level sets by the [[Liouville–Arnold theorem|PNQCW85HTJ]].
They are the natural coordinates for perturbation theory, such as [[KAM theory|3ZZSXKM16Z]].

## Explanations

### L1 human

Think of a motion that winds around a doughnut.
The actions say which doughnut you are on, and they never change.
The angles say where you are on that doughnut, and they increase at steady rates.
For the harmonic oscillator $H = \frac{1}{2}(p^2 + q^2)$, the action is the energy and the angle is the phase of the oscillation, which grows linearly in time.