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  "body": "\n## Summary\n\nA Hamilton–Jacobi equation is a first-order nonlinear [[partial differential equation|GC5BDZ6QBC]] of the form $u_t + H(\\nabla u, x) = 0$, where $H$ is a given function called the Hamiltonian.\nIt comes from classical mechanics, where its solutions generate the motions of a [[Hamiltonian system|0EDKGA2FAK]].\nIt also appears in optimal control, geometric optics and front propagation.\nClassical solutions usually stop existing after some time, and the right notion of global solution is the [[viscosity solution|H1ERJ3T75T]].\nSee the [[definition|4H1E4KW1JX]].\n\n## Explanations\n\n### L1 human\n\nImagine many travellers who all want to reach a goal as cheaply as possible.\nLet $u(x, t)$ be the least cost of a journey that ends at the point $x$ at time $t$.\nThe Hamilton–Jacobi equation is the rule that this least cost must obey when one moves $x$ or $t$ a little.\nFor $H(p) = \\abs{p}^2/2$ the equation reads $u_t + \\tfrac12 \\abs{\\nabla u}^2 = 0$, and for example $u(x, t) = \\abs{x}^2 / (2(t + 1))$ is a solution for $t \u003e -1$.\n",
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