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  "body": "\n## Statement\n\nLet $(M, \\omega)$ be a [[symplectic manifold|CAN490EPRT]] of dimension $2n$.\nLet $F_1, \\ldots, F_n \\colon M \\to \\R$ be smooth functions with [[Poisson brackets|2S0MXDT4YH]] $\\{F_i, F_j\\} = 0$ for all $i, j$, and let $H = F_1$.\nLet $F = (F_1, \\ldots, F_n) \\colon M \\to \\R^n$, let $c \\in \\R^n$ be a regular value of $F$, and let $M_c = F^{-1}(c)$.\nThen:\n\n1. $M_c$ is a smooth [[Lagrangian submanifold|H8ACPTMX08]] of dimension $n$. It is invariant under the flows of the [[Hamiltonian vector fields|C5EH3B4BEP]] $X_{F_1}, \\ldots, X_{F_n}$.\n2. Each compact connected component of $M_c$ is diffeomorphic to the [[torus|03NM7DZXXC]] $\\mathbb{T}^n = \\R^n / 2\\pi \\Z^n$.\n3. On such a component there are angle coordinates $\\theta \\in \\mathbb{T}^n$ in which the flow of $X_H$ is linear: $\\theta(t) = \\theta(0) + t \\, \\nu(c)$ for a vector $\\nu(c) \\in \\R^n$.\n4. A neighbourhood of such a component has [[action-angle coordinates|M3T81YNSKC]] $(I, \\theta)$ in which $H$ is a function of $I$ only.\n\n## Remarks\n\nSince the $F_i$ Poisson-commute, the flows of $X_{F_1}, \\ldots, X_{F_n}$ commute and preserve $M_c$.\nThey give an action of $\\R^n$ on each component of $M_c$, which is locally free because the $dF_i$ are independent there.\nA compact connected manifold with a transitive locally free $\\R^n$-action is $\\R^n / \\Lambda$ for a lattice $\\Lambda$, that is, a torus.\nThe actions are built from the integrals $\\oint p \\, dq$ over the basic cycles of the tori.\n\nExample: for two uncoupled harmonic oscillators on $\\R^4$, with $F_1 = \\frac{1}{2}(p_1^2 + q_1^2) + \\frac{1}{2}(p_2^2 + q_2^2)$ and $F_2 = \\frac{1}{2}(p_2^2 + q_2^2)$, the level sets with $c_1 \u003e c_2 \u003e 0$ are products of two circles, that is, $2$-tori.\n\nCompactness is necessary for the torus conclusion.\nFor a free particle on $\\R$ with $H = p^2/2$, the level set $H = c \u003e 0$ is two lines, not circles, and the motion is not periodic.\n\n## Explanations\n\n### L1 human\n\nIf a mechanical system with $n$ degrees of freedom has $n$ independent conserved quantities that are compatible with each other, then its motion is very simple.\nEach bounded motion lives on a doughnut-shaped surface, an $n$-dimensional torus.\nOn the torus, the motion is a straight line wound around at constant speeds.\nThis is one precise meaning of \"the system can be solved\".\n\n### L2 human\n\nThe theorem is the structure theorem for [[integrable systems|N0N7CMRCMH]].\nIt turns the algebraic condition of commuting integrals into a geometric picture: a fibration of phase space by invariant Lagrangian tori, with linear flow on each torus.\nThe frequency vector $\\nu$ decides the type of motion.\nWhen its entries are rationally independent, each orbit is dense in its torus.\nWhen they satisfy an integer relation, the torus is filled by invariant tori of lower dimension, and no orbit is dense in it.\nThis picture is the starting point of [[KAM theory|3ZZSXKM16Z]], which asks which tori survive a small perturbation of $H$.\n",
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