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  "body": "\n## Summary\n\nThe Weierstrass excess function of a Lagrangian $L(x, y, p)$ is $E(x, y, p, q) = L(x, y, q) - L(x, y, p) - (q - p) \\cdot L_p(x, y, p)$.\nIt measures how far the graph of $q \\mapsto L(x, y, q)$ lies above its tangent at $q = p$.\nIt gives the Weierstrass necessary condition for a strong minimum, and through Hilbert's invariant integral it gives the classical sufficient condition for a strong minimum.\n\n## Explanations\n\n### L1 human\n\nFix a point and a slope $p$ of a curve, and compare the integrand at another slope $q$ with its tangent-line prediction from $p$.\nThe excess is the gap between the true value and the prediction.\nIf the integrand is convex in the slope, the gap is never negative.\nFor $L = p^2$ the excess is $q^2 - p^2 - 2p(q - p) = (q - p)^2$.\nAlong a strong minimiser the excess must be nonnegative; otherwise a short sudden switch to the slope $q$ would lower the integral.\n",
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