Hill differential equation

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 and electromagnetic structures that are periodic in space[6] and/or in time.[7]

  1. ^ a b Magnus, W.; Winkler, S. (2013). Hill's equation. Courier. ISBN 9780486150291.
  2. ^ a b 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.
  4. ^ Sheretov, Ernst P. (April 2000). "Opportunities for optimization of the rf signal applied to electrodes of quadrupole mass spectrometers.: Part I. General theory". International Journal of Mass Spectrometry. 198 (1–2): 83–96. doi:10.1016/S1387-3806(00)00165-2.
  5. ^ Casperson, Lee W. (November 1984). "Solvable Hill equation". Physical Review A. 30: 2749. doi:10.1103/PhysRevA.30.2749.
  6. ^ Brillouin, L. (1946). Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices, McGraw–Hill, New York
  7. ^ Koutserimpas, Theodoros T.; Fleury, Romain (October 2018). "Electromagnetic Waves in a Time Periodic Medium With Step-Varying Refractive Index". IEEE Transactions on Antennas and Propagation. 66 (10): 5300–5307. doi:10.1109/TAP.2018.2858200.