Basis (universal algebra): Difference between revisions

Revision as of 23:56, 9 January 2014

This article is about the Hill differential equation; for the equation used in biochemistry see Hill equation (biochemistry)

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

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

where f(t) is a periodic function.^[1] It is named after George William Hill, who introduced it in 1886.^[2]

One can always assume that the period of f(t) equals 2π; then the Hill equation can be rewritten using the Fourier series of f(t):

\frac{d^{2} y}{d t^{2}} + (θ_{0} + 2 \sum_{n = 1}^{\infty} θ_{n} \cos (2 n t)) 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 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.

Aside from its original application to lunar stability, the Hill equation appears in many settings including the modeling of a quadrupole mass spectrometer, as the one-dimensional Schrödinger equation of an electron in a crystal and in accelerator physics.

References

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[2]

[3]

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+:''This article is about the Hill differential equation; for the equation used in biochemistry see [[Hill equation (biochemistry)]]''
+In [[mathematics]], the '''Hill equation''' or '''Hill differential equation''' is the second-order linear [[ordinary differential equation]]
+:<math> \frac{d^2y}{dt^2} + f(t) y = 0, </math>
+where ''f(t)'' is a [[periodic function]].<ref>{{cite book|last=Magnus|first=W.|last2=Winkler|first2=S.|title=Hill's equation|year=1966|publisher=Interscience Publishers John Wiley & Sons|location=New York-London-Sydney}}</ref> It is named after [[George William Hill]], who introduced it in 1886.<ref>{{cite journal|doi=10.1007/BF02417081|authorlink=George William Hill|first=G.W.|last=Hill|title=On the Part of the Motion of Lunar Perigee Which is a Function of the Mean Motions of the Sun and Moon|journal=Acta Math.|volume=8|issue=1|pages=1–36|year=1886}}</ref>
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+One can always assume that the period of ''f''(''t'') equals 2''π''; then the Hill equation can be rewritten using the [[Fourier series]] of ''f''(''t''):
+:<math>\frac{d^2y}{dt^2}+\left(\theta_0+2\sum_{n=1}^\infty \theta_n \cos(2nt) \right ) y=0. </math>
+Important special cases of Hill's equation include the [[Mathieu_function#Mathieu_equation|Mathieu equation]] (in which only the terms corresponding to ''n''&nbsp;=&nbsp;0,&nbsp;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 ''f''(''t''), solutions may stay bounded for all time, or the amplitude of the oscillations in solutions may grow exponentially.<ref>{{cite book
+ | last = Teschl
+ | given = Gerald
+ |authorlink=Gerald Teschl
+ | title = Ordinary Differential Equations and Dynamical Systems
+ | publisher=[[American Mathematical Society]]
+ | place = [[Providence, Rhode Island|Providence]]
+ | year = 2012
+ | isbn= 978-0-8218-8328-0
+ | url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}</ref> 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 the modeling of a [[quadrupole mass spectrometer]], as the one-dimensional [[Schrödinger equation]] of an electron in a crystal and in [[Accelerator_physics#Beam_dynamics|accelerator physics]].
+==References==
+{{Reflist}}
+<!---
+* [[Heinrich Guggenheimer]] (1977) ''Applicable Geometry'', pages 73&ndash;98, Krieger, Huntington ISBN 0-88275-368-1 .
+-->
+==External links==
+* {{springer|title=Hill equation|id=p/h047440}}
+* {{mathworld|urlname=HillsDifferentialEquation|title=Hill's Differential Equation}}
+*{{dlmf|first=G.|last=Wolf|id=28.29|title=Mathieu Functions and Hill’s Equation}}
+[[Category:Ordinary differential equations]]
+{{mathapplied-stub}}

Basis (universal algebra): Difference between revisions

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