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  "body": "\n## Statement\n\nLet $a \u003c b$, and let $L \\colon [a, b] \\times \\R^n \\times \\R^n \\to \\R$, written $L(x, y, p)$, be of class $C^2$.\nLet $J(y) = \\int_a^b L(x, y(x), y'(x)) \\, dx$ be the [[integral functional|JRQZ803Q5Y]] on $C^1([a, b]; \\R^n)$.\nLet $y_0 \\in C^1([a, b]; \\R^n)$ be a [[weak local minimiser|WMPCR723MJ]] of $J$ among functions with the same values at $a$ and $b$.\nThen for every $x \\in [a, b]$ the $n \\times n$ matrix $L_{pp}(x, y_0(x), y_0'(x))$ of second derivatives in $p$ is [[positive semidefinite|01MMVW8CKC]].\nFor $n = 1$ this reads $L_{pp}(x, y_0(x), y_0'(x)) \\ge 0$.\n\n## Remarks\n\nThe [[second variation|4C3BANJ8AH]] at a minimiser is nonnegative: $\\int_a^b \\big( h'^T L_{pp} h' + 2 h^T L_{yp} h' + h^T L_{yy} h \\big) \\, dx \\ge 0$ for all $h \\in C^1$ with $h(a) = h(b) = 0$.\nIf $L_{pp}$ had a negative direction $v$ at some interior point $x_0$, take $h$ to be a small zigzag in direction $v$ near $x_0$, with slopes $\\pm 1$ and tiny height.\nThen the term with $h'^2$ dominates the others, because $h$ itself is small, and the second variation becomes negative.\nThe endpoints follow by continuity.\n\nExample: for $J(y) = \\int_0^1 (1 - y'^2) \\, dx$ one has $L_{pp} = -2 \u003c 0$ everywhere, so $J$ has no weak local minimiser with fixed end values.\n\nFor a maximiser the matrix is negative semidefinite.\nThe condition is only necessary: the extremal $y_0 = 0$ of $\\int_0^{2\\pi} (y'^2 - y^2) \\, dx$ satisfies $L_{pp} = 2 \u003e 0$ but is not a minimiser.\nThe *strengthened Legendre condition* asks that $L_{pp}$ be positive definite; together with the Jacobi condition it gives a sufficient condition for a weak minimum.\n\n## Explanations\n\n### L1 human\n\nAt a minimum of an ordinary function, the second derivative cannot be negative.\nThe Legendre condition is the analogue for curves that minimise an integral.\nIt says that the integrand must curve upward as a function of the slope, along the minimising curve.\n\n### L2 human\n\nThe Legendre condition is the second necessary condition of the classical theory, after the Euler–Lagrange equation.\nIt only involves the highest-order part of the second variation, the coefficient of $h'^2$.\nThe full sign of the second variation is controlled by the Jacobi equation and conjugate points.\nIn several variables its analogue is the Legendre–Hadamard condition, and in optimal control it becomes the Legendre–Clebsch condition on the Hamiltonian.\n",
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