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  "body": "\n## Summary\n\nA natural boundary condition is a boundary condition that a minimiser satisfies automatically at an end where no value is prescribed.\nFor $J(y) = \\int_a^b L(x, y, y') \\, dx$ with $y(b)$ free, it reads $L_p = 0$ at $x = b$.\nIt comes from the [[first variation|5B8NFYB2XQ]], not from the statement of the problem.\nFor the Dirichlet energy in several variables, it is the [[Neumann boundary condition|GS3KT2H645]].\n\n## Explanations\n\n### L1 human\n\nIf one end of a curve is free to move up and down, the best curve has to be best also with respect to that freedom.\nThis gives an extra condition at the free end, which nobody imposed by hand.\nThink of a hanging string with one end on a frictionless vertical rail.\nThe rail cannot push along the string, so the string settles with its end meeting the rail at a right angle.\n",
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