Hill differential equation

In mathematics, the Hill equation or Hill differential equation is the second-order linear ordinary differential equation

${\displaystyle {\frac {d^{2}y}{dt^{2}}}+f(t)y=0,}$

where ${\displaystyle f(t)}$ is a periodic function with minimal period ${\displaystyle \pi }$ and average zero. By these we mean that for all ${\displaystyle t}$

${\displaystyle f(t+\pi )=f(t),}$

and

${\displaystyle \int _{0}^{\pi }f(t)\,dt=0,}$

and if ${\displaystyle p}$ is a number with ${\displaystyle 0<p<\pi }$, the equation ${\displaystyle f(t+p)=f(t)}$ must fail for some ${\displaystyle t}$.^[1] It is named after George William Hill, who introduced it in 1886.^[2]

Because ${\displaystyle f(t)}$ has period ${\displaystyle \pi }$, the Hill equation can be rewritten using the Fourier series of ${\displaystyle f(t)}$:

${\displaystyle {\frac {d^{2}y}{dt^{2}}}+\left(\theta _{0}+2\sum _{n=1}^{\infty }\theta _{n}\cos(2nt)+\sum _{m=1}^{\infty }\phi _{m}\sin(2mt)\right)y=0.}$

Important special cases of Hill's equation include the Mathieu equation (in which only the terms corresponding to n = 0, 1 are included) and the Meissner equation.

Hill's equation is an important example in the understanding of periodic differential equations. Depending on the exact shape of ${\displaystyle f(t)}$, solutions may stay bounded for all time, or the amplitude of the oscillations in solutions may grow exponentially.^[3] The precise form of the solutions to Hill's equation is described by Floquet theory. Solutions can also be written in terms of Hill determinants.^[1]

Aside from its original application to lunar stability,^[2] the Hill equation appears in many settings including in modeling of a quadrupole mass spectrometer,^[4] as the one-dimensional Schrödinger equation of an electron in a crystal,^[5] quantum optics of two-level systems, accelerator physics and electromagnetic structures that are periodic in space^[6] and/or in time.^[7]