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  "body": "\n## Summary\n\nThe Euler–Lagrange equation of an integral functional $\\int L(x, u, \\nabla u) \\, dx$ is the differential equation satisfied by its smooth critical points, in particular by its smooth minimisers.\nIt is the analogue, for functionals, of the condition that the derivative vanishes at a minimum.\nMany equations of physics and geometry arise this way.\nExamples are Laplace's equation from the Dirichlet energy, the minimal surface equation from area, and Newton's equations from the action.\nIt is the basic necessary condition of the [[calculus of variations|PQ25NHV0KS]].\n\n## Explanations\n\n### L1 human\n\nIf a function makes an energy as small as possible, then small changes of the function cannot lower the energy, to first order.\nWriting out this condition and integrating by parts gives a differential equation, the Euler–Lagrange equation.\nFor the length $\\int \\sqrt{1 + y'^2} \\, dx$ of a graph, the equation says $y'' = 0$, so shortest graphs are straight lines.\nFor the energy $\\int \\frac{1}{2} \\abs{\\nabla u}^2 \\, dx$, the equation is Laplace's equation $\\Delta u = 0$.\n",
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