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  "body": "\n## Problem\n\nLet $A$ and $B$ be two points in a vertical plane, with $B$ lower than $A$ and not on the vertical line through $A$.\nA particle starts at rest at $A$ and slides without friction along a curve from $A$ to $B$ under constant gravity $g$.\nWhich curve makes the time of travel from $A$ to $B$ as small as possible?\n\nIn coordinates, put $A = (0, 0)$, measure $y$ downwards, and let $B = (x_1, y_1)$ with $x_1, y_1 \u003e 0$.\nEnergy conservation gives the speed $\\sqrt{2 g y}$.\nSo for a graph $y(x)$ the time is the [[integral functional|JRQZ803Q5Y]]\n$$T(y) = \\int_0^{x_1} \\sqrt{\\frac{1 + y'(x)^2}{2 g \\, y(x)}} \\, dx,$$\nto be minimised among curves with $y(0) = 0$ and $y(x_1) = y_1$.\n\n## Remarks\n\nThe answer is an arc of a cycloid, the curve traced by a point on a circle that rolls along a horizontal line.\nIts parametric form is $x = r(\\theta - \\sin \\theta)$, $y = r(1 - \\cos \\theta)$, with the radius $r$ fixed by the condition that the curve passes through $B$.\nThe integrand does not depend on $x$, so the [[Beltrami identity|F7EKTEESQ5]] applies and gives $y (1 + y'^2) = 2r$ for a constant $r$; the cycloid solves this equation.\nThe fastest curve is not the straight segment; when $x_1 / y_1 \u003e \\pi / 2$, the cycloid even dips below the height of $B$ before it rises to $B$.\nJohann Bernoulli posed the problem as a public challenge in Acta Eruditorum; solutions by Johann Bernoulli, Jacob Bernoulli, Leibniz and Newton appeared in 1697.\nThe problem is often taken as the start of the calculus of variations.\n\n## Explanations\n\n### L1 human\n\nA ball rolls from a high point to a lower point off to the side, and one wants the track that gets it there fastest.\nThe straight track is the shortest but not the fastest.\nA track that first drops steeply lets the ball pick up speed early, and the best balance between length and speed is a cycloid.\n",
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