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,
References
- ^ ISBN 9780486150291.
- ^ .
- ISBN 978-0-8218-8328-0.
- .
- .
- ^ Brillouin, L. (1946). Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices, McGraw–Hill, New York
- .
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 MR 2723248.