Hill differential equation
In mathematics, the Hill equation or Hill differential equation is the second-order linear ordinary differential equation
- [math]\displaystyle{ \frac{d^2y}{dt^2} + f(t) y = 0, }[/math]
where [math]\displaystyle{ f(t) }[/math] is a periodic function by minimal period [math]\displaystyle{ \pi }[/math]. By these we mean that for all [math]\displaystyle{ t }[/math]
- [math]\displaystyle{ f(t+\pi)=f(t), }[/math]
and
- [math]\displaystyle{ \int_0^\pi f(t) \,dt=0, }[/math]
and if [math]\displaystyle{ p }[/math] is a number with [math]\displaystyle{ 0 \lt p \lt \pi }[/math], the equation [math]\displaystyle{ f(t+p) = f(t) }[/math] must fail for some [math]\displaystyle{ t }[/math].[1] It is named after George William Hill, who introduced it in 1886.[2]
Because [math]\displaystyle{ f(t) }[/math] has period [math]\displaystyle{ \pi }[/math], the Hill equation can be rewritten using the Fourier series of [math]\displaystyle{ f(t) }[/math]:
- [math]\displaystyle{ \frac{d^2y}{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. }[/math]
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 [math]\displaystyle{ f(t) }[/math], 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]
References
- ↑ 1.0 1.1 Magnus, W.; Winkler, S. (2013). Hill's equation. Courier. ISBN 9780486150291.
- ↑ 2.0 2.1 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. https://zenodo.org/record/1691491.
- ↑ Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0. https://www.mat.univie.ac.at/~gerald/ftp/book-ode/.
- ↑ 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.
- ↑ Casperson, Lee W. (November 1984). "Solvable Hill equation". Physical Review A 30: 2749. doi:10.1103/PhysRevA.30.2749.
- ↑ Brillouin, L. (1946). Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices, McGraw–Hill, New York
- ↑ 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.
External links
- Hazewinkel, Michiel, ed. (2001), "Hill equation", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/h047440
- Weisstein, Eric W.. "Hill's Differential Equation". http://mathworld.wolfram.com/HillsDifferentialEquation.html.
- Wolf, G. (2010), "Mathieu Functions and Hill’s Equation", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/28.29
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