Hill differential equation

This article is about the Hill differential equation; for the equation used in biochemistry see Hill equation (biochemistry)

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

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

where f(t) is a periodic function.^[1] It is named after George William Hill, who introduced it in 1886.^[2]

One can always assume that the period of f(t) equals π; then the Hill equation can be rewritten using the Fourier series of f(t):

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

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 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.

Aside from its original application to lunar stability, the Hill equation appears in many settings including the modeling of a quadrupole mass spectrometer, as the one-dimensional Schrödinger equation of an electron in a crystal and in accelerator physics.

References

1. Magnus, W.; Winkler, S. (2013). Hill's equation. Courier. ISBN 9780486150291.
2. Hill, G.W. (1886). "On the Part of the Motion of Lunar Perigee Which is a Function of the Mean Motions of the Sun and Moon". Acta Math. 8 (1): 1–36. doi:10.1007/BF02417081.
3. Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.

External links

• Hazewinkel, Michiel, ed. (2001), "Hill equation", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Weisstein, Eric W. "Hill's Differential Equation". MathWorld.

• Wolf, G. (2010), "Mathieu Functions and Hill's Equation", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248