In mathematics, the Hill equation or Hill differential equation is the second-order linear ordinary differential equation
where is a periodic function with minimal period and average zero. By these we mean that for all
and
and if is a number with , the equation must fail for some . [1] It is named after George William Hill, who introduced it in 1886. [2]
Because has period , the Hill equation can be rewritten using the Fourier series of :
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 , 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 [6] and electromagnetic structures that are periodic in space [7] and/or in time. [8]