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  "body": "\n## Example\n\nLet $m \u003e 0$, $k \u003e 0$ and $c \u003e 0$ be constants. The damped harmonic oscillator is the equation\n$$m\\, x'' + c\\, x' + k\\, x = 0, \\quad \\text{that is} \\quad x'' + 2 \\zeta \\omega_0\\, x' + \\omega_0^2\\, x = 0,$$\nwith $\\omega_0 = \\sqrt{k/m}$ and damping ratio $\\zeta = c / \\left(2 \\sqrt{m k}\\right)$.\nThe characteristic roots are $r = \\omega_0 \\left(-\\zeta \\pm \\sqrt{\\zeta^2 - 1}\\right)$, and there are three cases.\nIf $\\zeta \u003e 1$ (overdamped), the roots are real, distinct and negative, and $x = c_1 e^{r_1 t} + c_2 e^{r_2 t}$.\nIf $\\zeta = 1$ (critically damped), $r = -\\omega_0$ is a double root, and $x = (c_1 + c_2 t) e^{-\\omega_0 t}$.\nIf $0 \u003c \\zeta \u003c 1$ ([[underdamped|0JJ6GDDE7Z]]), $x = e^{-\\zeta \\omega_0 t} \\left(c_1 \\cos \\omega_1 t + c_2 \\sin \\omega_1 t\\right)$ with $\\omega_1 = \\omega_0 \\sqrt{1 - \\zeta^2}$.\n\n## Remarks\n\nIn each case, each solution tends to $0$ as $t \\to \\infty$. The origin is an asymptotically stable [[equilibrium point|0N8NGKDZDZ]] of the system for $(x, x')$.\nThe term $c x'$ models a friction force proportional to the velocity.\nThe energy $E = \\frac{1}{2} m (x')^2 + \\frac{1}{2} k x^2$ satisfies $E' = -c (x')^2 \\le 0$. So the energy decreases along each solution.\n\n## Explanations\n\n### L1 human\n\nFriction takes energy from the oscillating mass.\nWith little friction, the mass still oscillates, but the swings get smaller.\nWith much friction, the mass creeps back to rest without any swing.\n",
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