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  "body": "\n## Summary\n\nIn the calculus of variations a local minimum depends on what \"nearby\" means for functions.\nA strong local minimum beats every admissible function whose values are close.\nA weak local minimum only beats those whose values and slopes are both close.\nEvery strong local extremum is a weak one, but not conversely.\nThe [[Euler–Lagrange equation|PS0VV149ET]] and the [[Legendre condition|4HZX2EN319]] are necessary for weak extrema, while the [[Weierstrass excess function|R8TWRPSPNV]] gives a further necessary condition for strong ones.\n\n## Explanations\n\n### L1 human\n\nTwo curves can be close in height but very different in slope, like a straight line and a fine zigzag near it.\nA weak minimum only has to win against curves that are close in height and in slope.\nA strong minimum must also win against wiggly curves that stay close only in height.\nTake the functional $\\int_0^1 y'(x)^3 \\, dx$ with ends $y(0) = 0$ and $y(1) = 1$.\nThe line $y = x$ wins against all gentle perturbations.\nBut a curve that drops steeply for a very short time and then climbs gives a much lower value.\n",
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