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  "body": "\n## Definition\n\nLet $U$ be an open subset of $\\mathbb{R}^n$ and let $f : U \\to \\mathbb{R}^n$ be a function.\nLet $H : U \\to \\mathbb{R}$ be a continuously differentiable function. It is a first integral of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$ if\n$$\\nabla H(x) \\cdot f(x) = 0 \\quad \\text{for each } x \\in U,$$\nand if $H$ is not constant on any non-empty open subset of $U$.\n\n## Remarks\n\nThe condition means that $\\frac{d}{dt} H(\\gamma(t)) = 0$ for each [[solution|04SW7WECS8]] $\\gamma$. So $H$ is constant along each solution.\nEach [[trajectory|0BVT6VSB3S]] lies in a level set $\\{x : H(x) = c\\}$. In the plane, the level curves of $H$ give the phase portrait.\nFor the [[harmonic oscillator|0YG0ZJPTWG]] $x' = y$, $y' = -x$, the function $H(x, y) = x^2 + y^2$ is a first integral.\nThe Hamiltonian of a time-independent [[Hamiltonian system|0C2AKPDCD9]] is a first integral.\n\n## Explanations\n\n### L1 human\n\nA first integral is a quantity that does not change while the system moves, such as the energy of a system without friction.\nKnowing it tells us that each motion stays on one level set of this quantity.\n",
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