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  "body": "\n## Summary\n\nAn integral functional assigns to a function $u$ the integral of an expression $L(x, u, \\nabla u)$ built from the point, the value of $u$ and its derivatives.\nThe function $L$ is the Lagrangian density.\nMost problems of the [[calculus of variations|PQ25NHV0KS]] concern such functionals, for example length, area, energy and action.\nTheir smooth critical points solve the [[Euler–Lagrange equation|PS0VV149ET]].\n\n## Explanations\n\n### L1 human\n\nMany quantities are computed by adding up small local contributions.\nThe length of a graph adds up small pieces $\\sqrt{1 + y'(x)^2} \\, dx$, and an energy adds up local energy densities.\nThe rule that gives the local contribution from the position, the value and the slope is the Lagrangian density $L$.\nFor $L(x, z, p) = \\sqrt{1 + p^2}$ on $[0, 1]$, the function $y(x) = x$ gets the value $\\int_0^1 \\sqrt{2} \\, dx = \\sqrt{2}$, the length of the diagonal of the unit square.\n",
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