Hill differential equation

Last updated

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]

References

  1. 1 2 Magnus, W.; Winkler, S. (2013). Hill's equation. Courier. ISBN   9780486150291.
  2. 1 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.
  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. Wiedemann, Helmut (1993). Particle Accelerator Physics: Basic Principles and Linear Beam Dynamics. Springer-Verlag Berlin Heidelberg GmbH. p. 196.
  7. Brillouin, L. (1946). Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices, McGraw–Hill, New York
  8. 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 .