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  "body": "\n## Definition\n\nLet $a \u003c b$ and let $L \\colon [a, b] \\times \\R \\times \\R \\to \\R$, written $L(x, y, p)$, be of class $C^3$.\nLet $J(y) = \\int_a^b L(x, y, y') \\, dx$ be the [[integral functional|JRQZ803Q5Y]] of $L$.\nLet $y_0 \\in C^2([a, b])$ be a solution of the [[Euler–Lagrange equation|PS0VV149ET]] of $J$.\nPut $P(x) = L_{pp}(x, y_0(x), y_0'(x))$ and $Q(x) = L_{yy}(x, y_0(x), y_0'(x)) - \\frac{d}{dx} L_{yp}(x, y_0(x), y_0'(x))$.\nThe *Jacobi equation* of $J$ along $y_0$ is the linear [[ordinary differential equation|0NT64JV96A]]\n$$-\\frac{d}{dx}\\big( P(x) \\, h'(x) \\big) + Q(x) \\, h(x) = 0$$\nfor an unknown function $h$ on $[a, b]$.\n\n## Remarks\n\nFor $h \\in C^1([a, b])$ with $h(a) = h(b) = 0$, the [[second variation|4C3BANJ8AH]] of $J$ at $y_0$ equals $\\int_a^b (P h'^2 + Q h^2) \\, dx$.\nThe Jacobi equation is the Euler–Lagrange equation of this quadratic functional.\nIt is also the linearisation of the Euler–Lagrange equation of $J$ at $y_0$.\nIf $y(x, c)$ is a smooth family of extremals with $y(x, 0) = y_0(x)$, then $h = \\partial y / \\partial c$ at $c = 0$ solves it.\n\nExample: for $L = p^2 - y^2$ and $y_0 = 0$, one has $P = 2$ and $Q = -2$.\nThe Jacobi equation is $h'' + h = 0$; the function $h = \\sin x$ satisfies it, since $h'' = -\\sin x = -h$.\n\nNon-example: for the same $L$, the function $h = x$ does not solve it, since $-2h'' - 2h = -2x \\ne 0$ for $x \\ne 0$.\nIt does solve the Jacobi equation $h'' = 0$ of $L = p^2$, whose extremals are straight lines.\n\nWhen $P \u003e 0$ on $[a, b]$, the equation has a unique solution for given $h(a)$ and $h'(a)$.\nFor curves $y \\colon [a, b] \\to \\R^n$ the same construction gives a linear system with matrices $P$ and $Q$.\n",
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