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  "body": "\n## Definition\n\nLet $X$ be a real vector space, let $U \\subseteq X$, and let $J \\colon U \\to \\R$ be a [[functional|1EQM0DJ4NC]].\nLet $y_0 \\in U$ and $h \\in X$ be such that $y_0 + \\varepsilon h \\in U$ for all small $\\abs{\\varepsilon}$.\nThe *second variation* of $J$ at $y_0$ in the direction $h$ is\n$$\\delta^2 J(y_0)[h] = \\frac{d^2}{d\\varepsilon^2} J(y_0 + \\varepsilon h) \\Big|_{\\varepsilon = 0},$$\nwhen this derivative exists.\n\nLet $J(y) = \\int_a^b L(x, y, y') \\, dx$ be an [[integral functional|JRQZ803Q5Y]] with $L(x, y, p)$ of class $C^2$.\nFor $y_0, h \\in C^1([a, b])$ the second variation is\n$$\\delta^2 J(y_0)[h] = \\int_a^b \\big( L_{pp} \\, h'^2 + 2 L_{yp} \\, h h' + L_{yy} \\, h^2 \\big) \\, dx,$$\nwhere the second partial derivatives of $L$ are taken at $(x, y_0(x), y_0'(x))$.\n\n## Remarks\n\nSome books put a factor $\\frac{1}{2}$ in front, so that $J(y_0 + \\varepsilon h) = J(y_0) + \\varepsilon \\, \\delta J(y_0)[h] + \\varepsilon^2 \\, \\delta^2 J(y_0)[h] + o(\\varepsilon^2)$, where $\\delta J$ is the [[first variation|5B8NFYB2XQ]].\n\nExample: let $J(y) = \\int_0^\\pi (y'^2 - y^2) \\, dx$ and $y_0 = 0$.\nThen $J(\\varepsilon h) = \\varepsilon^2 \\int_0^\\pi (h'^2 - h^2) \\, dx$, so $\\delta^2 J(0)[h] = 2 \\int_0^\\pi (h'^2 - h^2) \\, dx$.\nThis agrees with the formula, since $L_{pp} = 2$, $L_{yp} = 0$ and $L_{yy} = -2$.\n\nNon-example: let $J(y) = \\int_0^1 \\abs{y'} \\, dx$ on $C^1([0, 1])$ and $y_0 = 0$.\nThen $J(\\varepsilon h) = \\abs{\\varepsilon} \\int_0^1 \\abs{h'} \\, dx$, which is not differentiable at $\\varepsilon = 0$ when $h$ is not constant.\nSo the second variation at $0$ does not exist in such a direction $h$; the integrand $\\abs{p}$ is not of class $C^2$.\n\nIf $h(a) = h(b) = 0$ and $x \\mapsto L_{yp}(x, y_0(x), y_0'(x))$ is $C^1$, an integration by parts gives $\\delta^2 J(y_0)[h] = \\int_a^b \\big( P h'^2 + Q h^2 \\big) \\, dx$ with $P = L_{pp}$ and $Q = L_{yy} - \\frac{d}{dx} L_{yp}$.\n",
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