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  "body": "\n## Definition\n\nLet $V$ be a real [[vector space|000AYPA9AJ]], let $A \\subseteq V$, and let $J \\colon A \\to \\R$ be a [[functional|1EQM0DJ4NC]].\nLet $y \\in A$ and $\\eta \\in V$, and suppose that $y + t\\eta \\in A$ for all real $t$ with $\\abs{t}$ small enough.\nThe *first variation* of $J$ at $y$ in the direction $\\eta$ is\n$$\\delta J(y; \\eta) = \\frac{d}{dt} J(y + t\\eta) \\Big|_{t = 0} = \\lim_{t \\to 0} \\frac{J(y + t\\eta) - J(y)}{t},$$\nwhen this limit exists.\n\n## Remarks\n\nIt is also called the Gâteaux variation; it is a [[directional derivative|06KWGHDN2J]] of $J$.\nWhen the [[Gateaux derivative|EAZF9NJW12]] of $J$ at $y$ exists, $\\delta J(y; \\eta)$ is its value at $\\eta$.\nIf $V$ is normed and $J$ is Fréchet differentiable at $y$, then $\\delta J(y; \\eta) = DJ(y)\\eta$ for all $\\eta$.\n\nExample: on $V = C([0, 1])$ let $J(y) = \\int_0^1 y^2 \\, dx$.\nThen $J(y + t\\eta) = J(y) + 2t \\int_0^1 y\\eta \\, dx + t^2 \\int_0^1 \\eta^2 \\, dx$, so $\\delta J(y; \\eta) = 2 \\int_0^1 y \\eta \\, dx$.\n\nNon-example: for $J(y) = \\int_0^1 \\abs{y} \\, dx$, $y = 0$ and $\\eta = 1$, the quotient $(J(t\\eta) - J(0))/t = \\abs{t}/t$ has no limit as $t \\to 0$, so the first variation does not exist.\n\nFor an [[integral functional|JRQZ803Q5Y]] $J(y) = \\int_a^b L(x, y, y') \\, dx$ with $L$ of class $C^1$, and $y, \\eta \\in C^1([a, b])$, differentiation under the integral sign gives\n$$\\delta J(y; \\eta) = \\int_a^b \\big( L_z(x, y, y') \\, \\eta + L_p(x, y, y') \\, \\eta' \\big) \\, dx,$$\nwhere $L_z$ and $L_p$ are the partial derivatives of $L(x, z, p)$.\nIf $y$ is a local minimum of $J$ along the line $t \\mapsto y + t\\eta$ and $\\delta J(y; \\eta)$ exists, then $\\delta J(y; \\eta) = 0$.\n",
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