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  "body": "\n## Problem\n\nLet $\\ell \u003e 0$.\nConsider plane curves $\\gamma \\colon [0, \\ell] \\to \\R^2$ of class $C^2$, parametrised by arc length.\nPrescribe the end points $\\gamma(0)$, $\\gamma(\\ell)$ and the unit tangents $\\gamma'(0)$, $\\gamma'(\\ell)$.\nAmong such curves, find those that make the bending energy\n$$E(\\gamma) = \\frac{1}{2} \\int_0^\\ell \\kappa(s)^2 \\, ds$$\nstationary, where $\\kappa$ is the signed [[curvature|7JQS6VKFEZ]] of $\\gamma$.\nThe solutions are the *elasticae*, the shapes of a thin elastic rod held at its ends.\n\n## Remarks\n\nWrite $\\gamma'(s) = (\\cos \\theta(s), \\sin \\theta(s))$, so that $\\kappa = \\theta'$.\nLet $(X, Y) = \\gamma(\\ell) - \\gamma(0)$.\nThe end points give the constraints $\\int_0^\\ell \\cos \\theta \\, ds = X$ and $\\int_0^\\ell \\sin \\theta \\, ds = Y$.\nSo this is an [[isoperimetric problem|6FY0KHMK5A]] with two constraints.\nWith multipliers, the Euler–Lagrange equation becomes\n$$\\theta'' + c \\sin(\\theta - \\theta_0) = 0$$\nfor constants $c$ and $\\theta_0$, the equation of a [[pendulum|0SHPXM2HWH]].\nSo the solutions are expressed through elliptic integrals and [[elliptic functions|FF9ZPCZ59K]].\nEuler classified the possible shapes of the elasticae in 1744.\n\n## Explanations\n\n### L1 human\n\nBend a thin flexible ruler or a steel wire and hold its two ends.\nIt takes the shape that stores the least bending energy, which is measured by how sharply it curves along its length.\nThe angle of the wire along its length obeys the same equation as a swinging pendulum, and this gives a whole family of shapes, from gentle arcs to loops.\n",
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      "first_stated": "1691",
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