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  "body": "\n## Problem\n\nA bead slides without friction along a curve in a vertical plane, under a constant downward gravitational acceleration $g \u003e 0$, and starts at rest.\nFind the curves with a lowest point that have the following property.\nThe time the bead needs to reach the lowest point does not depend on its starting point on the curve.\n\n## Remarks\n\nThe problem is solved: the answer is an arc of a cycloid with its cusps pointing up.\nFor the cycloid traced by a circle of radius $a$, the descent time from every starting point is $\\pi \\sqrt{a / g}$.\nHuygens gave a geometric solution and used it for the design of pendulum clocks, in his book Horologium Oscillatorium of 1673.\n\nIn the nineteenth century Abel treated the problem through an integral equation.\nLet $y$ be the height above the lowest point and $s(y)$ the arc length from the lowest point, and let $\\varphi(y) = s'(y)$.\nBy conservation of energy, the speed at height $y$ of a bead released at height $h$ is $\\sqrt{2 g (h - y)}$, so the descent time is\n$$T(h) = \\frac{1}{\\sqrt{2g}} \\int_0^h \\frac{\\varphi(y)}{\\sqrt{h - y}} \\, dy.$$\nAsking that $T(h) = T$ for all $h$ is [[Abel's integral equation|G4GWBYTW7B]] with constant right side $\\sqrt{2g} \\, T$.\nThe [[Abel inversion formula|TDMXFY06A2]] gives $\\varphi(y) = \\frac{\\sqrt{2g} \\, T}{\\pi \\sqrt{y}}$, and the curves with $s'(y)$ proportional to $y^{-1/2}$ are exactly the cycloids, with $a = g T^2 / \\pi^2$.\nCheck: on the cycloid $s^2 = 8 a y$, energy conservation gives $s'' = -\\frac{g}{4a} s$, a harmonic oscillation whose quarter period $\\frac{\\pi}{2} \\sqrt{4a/g} = \\pi \\sqrt{a/g}$ does not depend on the amplitude.\n\n## Explanations\n\n### L1 human\n\nOn most slides, a ball released higher up takes longer to reach the bottom.\nOn a cycloid, the steeper upper part speeds up a ball released high just enough to compensate for the longer way.\nAll balls arrive at the bottom at the same moment, wherever they start.\nThe problem is a famous early example in which an unknown curve is found from a condition on travel times.\nAbel's treatment of it is often seen as a starting point of the theory of integral equations.\n",
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      "posed_by": "Christiaan Huygens"
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