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  "body": "\n## Example\n\nLet $m \u003e 0$ and $k \u003e 0$ be constants, and let $\\omega_0 = \\sqrt{k/m}$.\nThe equation of the harmonic oscillator is\n$$m\\, x''(t) + k\\, x(t) = 0, \\quad \\text{that is} \\quad x'' + \\omega_0^2\\, x = 0.$$\nIt models a mass $m$ on a spring with stiffness $k$, where $x$ is the displacement from the rest position.\nIts characteristic roots are $\\pm i \\omega_0$, so its solutions are $x(t) = c_1 \\cos(\\omega_0 t) + c_2 \\sin(\\omega_0 t) = A \\cos(\\omega_0 t + \\varphi)$ with constants $A \\ge 0$ and $\\varphi$.\n\n## Remarks\n\nEach nonzero solution is periodic with period $2 \\pi / \\omega_0$. The period does not depend on the amplitude $A$.\nThe energy $E = \\frac{1}{2} m (x')^2 + \\frac{1}{2} k x^2$ is constant along each solution. It is a [[first integral|0SS1P1YC9A]].\nWith friction, the equation becomes the [[damped harmonic oscillator|0RNXVVTJTJ]].\n\n## Explanations\n\n### L1 human\n\nThe spring pulls the mass back with a force proportional to the displacement.\nThe mass overshoots the rest position, and the motion repeats forever as a sine wave.\nA stiffer spring or a lighter mass gives a faster oscillation.\n",
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