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  "body": "\n## Definition\n\nLet $(M, \\omega)$ be a [[symplectic manifold|CAN490EPRT]] of dimension $2n$, let $H \\colon M \\to \\R$ be smooth, and let $U \\subseteq M$ be open.\nLet $\\mathbb{T}^n = \\R^n / 2\\pi \\Z^n$ be the $n$-[[torus|03NM7DZXXC]].\n*Action-angle coordinates* for $H$ on $U$ are given by a [[diffeomorphism|0KZZ70ZHXP]] $\\Psi \\colon U \\to D \\times \\mathbb{T}^n$, with $D \\subseteq \\R^n$ open, written $\\Psi = (I_1, \\ldots, I_n, \\theta_1, \\ldots, \\theta_n)$, such that:\n\n1. $\\omega = \\sum_{k=1}^n dI_k \\wedge d\\theta_k$ on $U$;\n2. $H = K(I_1, \\ldots, I_n)$ on $U$ for a smooth function $K \\colon D \\to \\R$.\n\nThe $I_k$ are the *actions* and the $\\theta_k$ are the *angles*.\n\n## Remarks\n\nHere the convention is $\\omega = \\sum_k dp_k \\wedge dq_k$ on $\\R^{2n}$, so that the actions play the role of momenta and the angles the role of positions.\nIn these coordinates the [[Hamiltonian system|0EDKGA2FAK]] reads $I_k' = 0$ and $\\theta_k' = \\partial K / \\partial I_k (I)$, so $\\theta(t) = \\theta(0) + t \\, \\nabla K(I)$.\n\nExample: the harmonic oscillator $H = \\frac{1}{2}(p^2 + q^2)$ on $\\R^2 \\setminus \\set{0}$.\nPut $q = \\sqrt{2I} \\sin \\theta$ and $p = \\sqrt{2I} \\cos \\theta$ with $I \u003e 0$.\nA short computation gives $dp \\wedge dq = dI \\wedge d\\theta$, and $H = I$, so both clauses hold with $K(I) = I$.\n\nNon-example: polar coordinates $q = r \\sin \\theta$, $p = r \\cos \\theta$ with $r \u003e 0$ give $H = r^2/2$, but $dp \\wedge dq = r \\, dr \\wedge d\\theta$, so clause 1 fails.\n\nFor an integrable system, the actions can be computed as $I_k = \\frac{1}{2\\pi} \\oint_{\\gamma_k} p \\, dq$ over basic cycles $\\gamma_k$ of an invariant torus, when $\\omega = d(\\sum_k p_k \\, dq_k)$.\n",
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