Hill differential equation

Not to be confused with Hill equation (biochemistry).

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 ${\displaystyle t}$.^[1] It is named after George William Hill, who introduced it in 1886.^[2]

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.^[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. 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 [1994]
• 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.