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  "body": "\n## Example\n\nLet $g \u003e 0$ be the acceleration of gravity and $\\ell \u003e 0$ the length of a rigid massless rod, with a point mass at its end. Let $\\theta(t)$ be the angle of the rod from the downward vertical at time $t$.\nWithout friction, the motion satisfies the pendulum equation\n$$\\theta''(t) + \\frac{g}{\\ell} \\sin \\theta(t) = 0.$$\nAs a first-order system, it is $\\theta' = \\omega$, $\\omega' = -\\frac{g}{\\ell} \\sin \\theta$. The function $E(\\theta, \\omega) = \\frac{1}{2} \\omega^2 - \\frac{g}{\\ell} \\cos \\theta$ is a [[first integral|0SS1P1YC9A]].\n\n## Remarks\n\nFor small angles, $\\sin \\theta \\approx \\theta$. This gives the [[harmonic oscillator|0YG0ZJPTWG]] $\\theta'' + \\frac{g}{\\ell} \\theta = 0$, with the solution $\\theta(t) = \\theta_0 \\cos\\left(\\sqrt{g/\\ell}\\, t\\right)$ for $\\theta(0) = \\theta_0$, $\\theta'(0) = 0$, and the period $2 \\pi \\sqrt{\\ell / g}$.\nFor the nonlinear equation, the period depends on the amplitude. It is given by an elliptic integral and it is longer than $2 \\pi \\sqrt{\\ell / g}$.\nThe equilibrium $(0, 0)$, hanging down, is stable. The equilibrium $(\\pi, 0)$, standing up, is a [[hyperbolic|0C5PREPZWG]] saddle, so it is unstable.\n\n## Explanations\n\n### L1 human\n\nGravity pulls the mass back toward the lowest point, with a force proportional to $\\sin \\theta$.\nFor small swings, this is nearly proportional to $\\theta$, and the pendulum behaves like a spring.\nFor large swings, the motion is slower, and with enough energy the pendulum turns over the top.\n",
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