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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^n$ be a bounded open set and let $L \\colon \\Omega \\times \\R \\times \\R^n \\to \\R$ be a continuous function, the *Lagrangian*, written $L(x, z, p)$.\nLet $\\mathcal{A}$ be a class of functions $u \\colon \\Omega \\to \\R$ with prescribed values $u = g$ on $\\partial\\Omega$, for which the integral below makes sense.\nFor example, $\\mathcal{A}$ may lie in $C^1(\\overline{\\Omega})$ or in a [[Sobolev space|M4CEZ5K9HP]] $W^{1,q}(\\Omega)$.\nThe *integral functional* with Lagrangian $L$ is\n$$J(u) = \\int_\\Omega L\\big(x, u(x), \\nabla u(x)\\big) \\, dx, \\qquad u \\in \\mathcal{A}.$$\nA *variational problem* asks for a minimiser of $J$ on $\\mathcal{A}$, or more generally for its critical points.\nA minimiser is a $u_0 \\in \\mathcal{A}$ with $J(u_0) \\le J(u)$ for all $u \\in \\mathcal{A}$.\nThe *calculus of variations* is the study of such problems: existence of minimisers, necessary and sufficient conditions for a minimum, and properties of minimisers.\n\n## Remarks\n\nThe same words are used for functionals with higher derivatives, for vector-valued $u$, and for constraints such as fixed volume or fixed $L^2$ norm.\n\nExample: for $n = 2$, the Dirichlet energy $L(x, z, p) = \\frac{1}{2}\\abs{p}^2$ gives $J(u) = \\frac{1}{2} \\int_\\Omega \\abs{\\nabla u}^2 \\, dx$.\nIts minimisers with boundary values $g$ are the harmonic functions with these boundary values, which is Dirichlet's principle.\n\nExample: the area of the graph of $u$ has Lagrangian $L(x, z, p) = \\sqrt{1 + \\abs{p}^2}$, and its critical points solve the [[minimal surface equation|3GV2E2YTZX]].\n\nNon-example: minimising $u \\mapsto \\max_{\\overline\\Omega} \\abs{u}$ is an optimisation over functions.\nBut this functional is not the integral of a Lagrangian, so it is not a variational problem in this sense.\n\n## Explanations\n\n### L1 human\n\nA variational problem asks for the function that makes an integral quantity, such as energy or area, as small as possible.\nThe calculus of variations is the theory of these problems.\n",
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