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  "body": "\n## Summary\n\nAction-angle coordinates are canonical coordinates $(I, \\theta)$ for an [[integrable system|N0N7CMRCMH]] in which the Hamiltonian depends on the actions $I$ only.\nIn these coordinates the actions are constant and the angles turn at constant speed, so the equations of motion are solved at once.\nThey exist near compact regular level sets by the [[Liouville–Arnold theorem|PNQCW85HTJ]].\nThey are the natural coordinates for perturbation theory, such as [[KAM theory|3ZZSXKM16Z]].\n\n## Explanations\n\n### L1 human\n\nThink of a motion that winds around a doughnut.\nThe actions say which doughnut you are on, and they never change.\nThe angles say where you are on that doughnut, and they increase at steady rates.\nFor 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.\n",
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