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  "body": "\n## Definition\n\nLet $a \u003c b$, let $n \\ge 1$ and let $L \\colon [a, b] \\times \\R^n \\times \\R^n \\to \\R$, written $L(x, y, p)$, be of class $C^1$.\nWrite $L_p$ for the vector of [[partial derivatives|05AXG3QRKR]] of $L$ in the variables $p$.\nThe *Weierstrass excess function* of $L$ is the function $E \\colon [a, b] \\times \\R^n \\times \\R^n \\times \\R^n \\to \\R$ given by\n$$E(x, y, p, q) = L(x, y, q) - L(x, y, p) - (q - p) \\cdot L_p(x, y, p).$$\n\n## Remarks\n\nExample: for $L(x, y, p) = \\sqrt{1 + p^2}$ with $n = 1$, the function $q \\mapsto L(x, y, q)$ is convex.\nA convex $C^1$ function lies above each of its tangent lines, so $E \\ge 0$ everywhere.\n\nNon-example of nonnegativity: for $L = (p^2 - 1)^2$ with $n = 1$, at $p = 0$ one gets $E(x, y, 0, q) = (q^2 - 1)^2 - 1$.\nThis is negative for $0 \u003c \\abs{q} \u003c \\sqrt{2}$, since $L$ is not convex in $p$.\n\nThe Weierstrass necessary condition reads as follows.\nLet $y_0 \\in C^1([a, b]; \\R^n)$ be a strong local minimiser of $J(y) = \\int_a^b L(x, y, y') \\, dx$ with fixed end values, see [[weak and strong extremum|WMPCR723MJ]].\nThen $E(x, y_0(x), y_0'(x), q) \\ge 0$ for all $x \\in [a, b]$ and all $q \\in \\R^n$.\n\nFor $C^2$ functions $L$, Taylor's formula gives $E(x, y, p, q) = \\frac{1}{2} (q - p)^T L_{pp}(x, y, \\bar p) (q - p)$ for some $\\bar p$ on the segment from $p$ to $q$.\nSo the Weierstrass condition implies the Legendre condition $L_{pp}(x, y_0, y_0') \\ge 0$.\n",
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