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  "body": "\n## Example\n\nLet $T \u003e 0$ and let $f : \\R \\to \\R$ be continuous and $T$-periodic.\n*Hill's equation* is the second-order [[linear differential equation|0PB7K8M7FG]]\n$$y'' + f(t)\\, y = 0.$$\nThe Mathieu equation $y'' + (a - 2q \\cos 2t)\\, y = 0$, with real constants $a$ and $q$, is the special case $f(t) = a - 2q\\cos 2t$, of period $\\pi$.\n\nFloquet analysis: let $y_1, y_2$ be the solutions with $y_1(0) = 1$, $y_1'(0) = 0$ and $y_2(0) = 0$, $y_2'(0) = 1$.\nThe monodromy matrix of the equivalent system for $(y, y')$ is $M$ with columns $(y_1(T), y_1'(T))$ and $(y_2(T), y_2'(T))$.\nThe system matrix has trace $0$, so by [[Liouville's formula|09PS3S0EF6]] $\\det M = 1$.\nHence the multipliers are the roots of $\\mu^2 - \\Delta \\mu + 1 = 0$, where $\\Delta = y_1(T) + y_2'(T)$ is the discriminant.\n\n1. If $\\abs{\\Delta} \u003c 2$, the multipliers are two distinct conjugate numbers of modulus $1$, and all solutions are bounded.\n2. If $\\abs{\\Delta} \u003e 2$, there is a real multiplier of modulus greater than $1$, and some solution is unbounded.\n\nCheck with constant $f = \\omega^2$, $\\omega \u003e 0$, viewed as $T$-periodic: $y_1 = \\cos \\omega t$ and $y_2 = \\sin(\\omega t)/\\omega$, so $\\Delta = 2\\cos \\omega T$ and $\\abs{\\Delta} \\le 2$, matching the bounded oscillations of the [[harmonic oscillator|0YG0ZJPTWG]].\n\n## Explanations\n\n### L1 human\n\nHill's equation describes an oscillator whose stiffness changes periodically in time.\nDepending on the parameters, the oscillation stays bounded or grows without limit; the growth is called parametric resonance.\nA single number $\\Delta$, computed from two solutions over one period, decides which case occurs.\nGeorge William Hill studied the equation for the motion of the Moon, and it also describes particles in periodic potentials.\n",
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