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

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

where ${\displaystyle f(t)}$ is a periodic function with minimal period ${\displaystyle \pi }$ and average zero. By these we mean that for all ${\displaystyle t}$

${\displaystyle f(t+\pi )=f(t),}$

and

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

and if ${\displaystyle p}$ is a number with ${\displaystyle 0<p<\pi }$, the equation ${\displaystyle f(t+p)=f(t)}$ must fail for some ${\displaystyle t}$. It is named after George William Hill, who introduced it in 1886.

Because ${\displaystyle f(t)}$ has period ${\displaystyle \pi }$, the Hill equation can be rewritten using the Fourier series of ${\displaystyle f(t)}$:

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

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 ${\displaystyle f(t)}$, solutions may stay bounded for all time, or the amplitude of the oscillations in solutions may grow exponentially. 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 in modeling of a quadrupole mass spectrometer, as the one-dimensional Schrödinger equation of an electron in a crystal, quantum optics of two-level systems, accelerator physics and electromagnetic structures that are periodic in space and/or in time.

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.
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.

External links

• "Hill equation", Encyclopedia of Mathematics, EMS Press, 2001
• 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. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.

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