Editing Celestial mechanics (section)

==Perturbation theory==
{{main|Perturbation theory}}
[[Perturbation theory]] comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly. (It is closely related to methods used in [[numerical analysis]], which [[Methods of computing square roots#Heron's method|are ancient]].) The earliest use of modern [[perturbation theory]] was to deal with the otherwise unsolvable mathematical problems of celestial mechanics: [[Isaac Newton|Newton]]'s solution for the orbit of the [[Moon]], which moves noticeably differently from a simple [[Kepler's laws of planetary motion|Keplerian ellipse]] because of the competing gravitation of the [[Earth]] and the [[Sun]].

[[Perturbation theory|Perturbation methods]] start with a simplified form of the original problem, which is carefully chosen to be exactly solvable. In celestial mechanics, this is usually a [[Kepler's laws of planetary motion|Keplerian ellipse]], which is correct when there are only two gravitating bodies (say, the [[Earth]] and the [[Moon]]), or a circular orbit, which is only correct in special cases of two-body motion, but is often close enough for practical use.

The solved, but simplified problem is then ''"perturbed"'' to make its [[differential equation|time-rate-of-change equations for the object's position]] closer to the values from the real problem, such as including the gravitational attraction of a third, more distant body (the [[Sun]]). The slight changes that result from the terms in the equations – which themselves may have been simplified yet again – are used as corrections to the original solution. Because simplifications are made at every step, the corrections are never perfect, but even one cycle of corrections often provides a remarkably better approximate solution to the real problem.

There is no requirement to stop at only one cycle of corrections. A partially corrected solution can be re-used as the new starting point for yet another cycle of perturbations and corrections. In principle, for most problems the recycling and refining of prior solutions to obtain a new generation of better solutions could continue indefinitely, to any desired finite degree of accuracy.

The common difficulty with the method is that the corrections usually progressively make the new solutions very much more complicated, so each cycle is much more difficult to manage than the previous cycle of corrections. [[Isaac Newton|Newton]] is reported to have said, regarding the problem of the [[Moon]]'s orbit ''"It causeth my head to ache."''<ref>{{Citation |last1=Cropper |first1=William H. |title=Great Physicists: The life and times of leading physicists from Galileo to Hawking |publisher=[[Oxford University Press]] |isbn=978-0-19-517324-6 |date=2004 |page=34}}.</ref>

This general procedure – starting with a simplified problem and gradually adding corrections that make the starting point of the corrected problem closer to the real situation – is a widely used mathematical tool in advanced sciences and engineering. It is the natural extension of the "guess, check, and fix" method [[Methods of computing square roots#Heron's method|used anciently with numbers]].