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  "body": "\n## Summary\n\nThe Jacobi equation of a variational problem is the linear second-order differential equation $-(P h')' + Q h = 0$ obtained from the [[second variation|4C3BANJ8AH]] along an extremal.\nIt is the [[Euler–Lagrange equation|PS0VV149ET]] of the quadratic functional $\\int (P h'^2 + Q h^2) \\, dx$, and also the linearisation of the Euler–Lagrange equation along the extremal.\nIts solutions describe how nearby extremals spread apart, and their zeros define conjugate points.\nFor the length functional on a Riemannian manifold it becomes the equation of a [[Jacobi field|9AHMM41PJ3]].\n\n## Explanations\n\n### L1 human\n\nTake an extremal curve and wiggle it slightly into a family of nearby extremals.\nThe small gap between the curve and its neighbours, to first order, solves the Jacobi equation.\nFor the functional $\\int (y'^2 - y^2) \\, dx$ the Jacobi equation is $h'' + h = 0$, with solutions $h = c \\sin(x - x_0)$.\nThese solutions vanish again after a distance $\\pi$, which is why this extremal stops being a minimiser on intervals longer than $\\pi$.\n",
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