Hill differential equation explained

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

(d ^{2y})/(dt ^{2}) + f(t) y = 0 ,

where

f(t) is a periodic function with minimal period \pi and average zero. By these we mean that for all t f(t + \pi) = f(t),

and

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

and if

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

Because

f(t) has period \pi , the Hill equation can be rewritten using the Fourier series of f(t) :(d ^{2y})/(dt ^{2}) + \left(\theta _{0+2\sum} ^{inf ty} ^{n =1} \theta _{n} \cos( 2 nt) + \sum ^{inf ty} _{m =1} \phi _{m} \sin( 2 mt) \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.^[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]

Notes and References

1. Book: Magnus. W.. Winkler. S.. Hill's equation. 2013. Courier. 9780486150291.
2. 10.1007/BF02417081. George William Hill. G.W.. Hill. 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. 1886. free.
3. Book: Teschl . Gerald Teschl

   . Gerald . Gerald Teschl . Ordinary Differential Equations and Dynamical Systems . . . 2012 . 978-0-8218-8328-0 .

4. Sheretov . Ernst P. . Opportunities for optimization of the rf signal applied to electrodes of quadrupole mass spectrometers.: Part I. General theory . . April 2000 . 198 . 1-2 . 83–96 . 10.1016/S1387-3806(00)00165-2.
5. Casperson . Lee W. . Lee Casperson . Solvable Hill equation . . November 1984 . 30 . 2749 . 10.1103/PhysRevA.30.2749.
6. [Léon Brillouin|Brillouin, L.]
7. Koutserimpas . Theodoros T. . Fleury . Romain . Electromagnetic Waves in a Time Periodic Medium With Step-Varying Refractive Index . . October 2018 . 66 . 10 . 5300–5307 . 10.1109/TAP.2018.2858200. free .