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  "body": "\n## Definition\n\nLet $(M, \\omega)$ be a [[symplectic manifold|CAN490EPRT]] of dimension $2n$, and let $H \\colon M \\to \\R$ be a smooth function.\nLet $\\{F, G\\}$ denote the [[Poisson bracket|2S0MXDT4YH]] of smooth functions $F, G$ on $M$.\nThe [[Hamiltonian system|0EDKGA2FAK]] with Hamiltonian $H$ is *integrable*, or *Liouville integrable*, if there are smooth functions $F_1 = H, F_2, \\ldots, F_n \\colon M \\to \\R$ such that:\n\n1. $\\{F_i, F_j\\} = 0$ for all $i, j \\in \\set{1, \\ldots, n}$;\n2. the differentials $dF_1, \\ldots, dF_n$ are linearly independent at each point of an open dense subset of $M$.\n\n## Remarks\n\nClause 1 with $i = 1$ says that each $F_j$ is a [[first integral|0262JP0NV9]] of the system, since $\\frac{d}{dt} F_j = \\{F_j, H\\}$ along solutions.\nFunctions that satisfy clause 1 are said to be in involution.\n\nExample: on $\\R^4$ with coordinates $(q_1, q_2, p_1, p_2)$, let $H = \\frac{1}{2}(p_1^2 + q_1^2) + \\frac{1}{2}(p_2^2 + q_2^2)$ and $F_2 = \\frac{1}{2}(p_2^2 + q_2^2)$.\nThen $\\{H, F_2\\} = \\{\\frac{1}{2}(p_2^2 + q_2^2), \\frac{1}{2}(p_2^2 + q_2^2)\\} = 0$, since the first summand of $H$ does not depend on $q_2, p_2$.\nThe differentials $dH$ and $dF_2$ are independent wherever $(q_1, p_1) \\ne (0, 0)$ and $(q_2, p_2) \\ne (0, 0)$, an open dense set.\n\nNon-example: the same $H$ with $F_2 = H$ satisfies clause 1 but not clause 2, since $dF_1 = dF_2$ everywhere.\n\nEvery Hamiltonian system with one degree of freedom ($n = 1$) is integrable as soon as $dH \\ne 0$ on an open dense set.\nThe [[Liouville–Arnold theorem|PNQCW85HTJ]] describes the motion of an integrable system near a compact regular level set.\n",
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