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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $U$ be an open subset of $\\mathbb{R}^n \\times \\mathbb{R}^n$ with coordinates $(q, p) = (q_1, \\ldots, q_n, p_1, \\ldots, p_n)$, and let $H : U \\to \\mathbb{R}$ be continuously differentiable.\nThe Hamiltonian system with Hamiltonian $H$ is the [[autonomous system|07BNRFQ0PJ]]\n$$q_i' = \\frac{\\partial H}{\\partial p_i}(q, p), \\qquad p_i' = -\\frac{\\partial H}{\\partial q_i}(q, p), \\qquad i = 1, \\ldots, n.$$\n\n## Remarks\n\nThe function $H$ is a [[first integral|0SS1P1YC9A]] when it is not constant on an open set: along a solution, $\\frac{d}{dt} H = \\sum_i \\left( \\frac{\\partial H}{\\partial q_i} q_i' + \\frac{\\partial H}{\\partial p_i} p_i' \\right) = 0$.\nFor $H(q, p) = \\frac{1}{2} p^2 + V(q)$ with $n = 1$, the system is $q' = p$, $p' = -V'(q)$. This is Newton's law $q'' = -V'(q)$ for a particle with potential energy $V$.\nThe [[harmonic oscillator|0YG0ZJPTWG]] has $V(q) = \\frac{1}{2} q^2$, and the [[pendulum|0SHPXM2HWH]] has $V(q) = -\\cos q$ in suitable units.\nA time-dependent Hamiltonian $H(t, q, p)$ gives a non-autonomous Hamiltonian system, and then $H$ need not be conserved.\n\n## Explanations\n\n### L1 human\n\nA Hamiltonian system describes a mechanical system without friction by one function, the total energy $H$.\nThe positions $q$ and the momenta $p$ change according to the partial derivatives of $H$, and the energy stays constant.\n",
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