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  "body": "\n## Definition\n\nLet $I \\subseteq \\R$ be an open interval and let $L \\colon I \\times \\R^n \\times \\R^n \\to \\R$, $(t, q, v) \\mapsto L(t, q, v)$, be of class $C^2$.\nAssume that for each $(t, q)$ the map $v \\mapsto \\partial L / \\partial v (t, q, v)$ is a diffeomorphism of $\\R^n$ and $\\partial^2 L / \\partial v^2$ is positive definite.\nThe variational problem is to find extremals of $\\int L(t, q, q') \\, dt$, the solutions of the [[Euler–Lagrange equation|PS0VV149ET]].\nIts Hamiltonian is $H(t, q, p) = p \\cdot v - L(t, q, v)$, where $v$ is the unique vector with $p = \\partial L / \\partial v(t, q, v)$.\nThe *Hamilton–Jacobi equation* of the problem is the [[partial differential equation|5TRXQKP7DY]]\n$$\\frac{\\partial S}{\\partial t} + H\\Big(t, q, \\frac{\\partial S}{\\partial q}\\Big) = 0$$\nfor a function $S$ of class $C^2$ on an open set $G \\subseteq I \\times \\R^n$.\n*Hamilton–Jacobi theory* is the study of the variational problem through the solutions $S$ of this equation and the curves $q$ in $G$ with\n$$q'(t) = \\frac{\\partial H}{\\partial p}\\Big(t, q(t), \\frac{\\partial S}{\\partial q}(t, q(t))\\Big).$$\n\n## Remarks\n\nThe basic fact is that each such curve is an extremal, with momentum $p(t) = \\partial S / \\partial q(t, q(t))$ solving Hamilton's canonical equations.\nThese curves fill $G$ and form a [[field of extremals|K4DFT3H9BJ]], and $S$ is the action along them.\nExample: $L = \\frac12 \\abs{v}^2$, so $H = \\frac12 \\abs{p}^2$, on $G = \\set{(t, q) : t \u003e t_0}$.\nThe function $S(t, q) = \\abs{q - a}^2 / (2(t - t_0))$ solves the equation, since $S_t = -\\abs{q - a}^2 / (2 (t - t_0)^2)$ and $\\frac12 \\abs{\\nabla_q S}^2 = \\abs{q - a}^2 / (2 (t - t_0)^2)$.\nThe curves solve $q' = (q - a)/(t - t_0)$, so they are the straight lines through $a$ at time $t_0$, which are the extremals of $L$.\nNon-example: for $L = \\frac12 \\abs{v}^2$ the function $S(t, q) = \\abs{q}^2$ does not solve the equation, since $S_t + \\frac12 \\abs{\\nabla S}^2 = 2 \\abs{q}^2 \\ne 0$ for $q \\ne 0$.\nJacobi's theorem completes the theory: from a family $S(t, q, a)$ of solutions with $\\det(\\partial^2 S / \\partial q \\, \\partial a) \\ne 0$, the equations $\\partial S / \\partial a = b$ define all solutions of Hamilton's equations near a point.\n",
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