Editing Perturbation theory (section)

==History==
Perturbation theory was first devised to solve [[Three-body problem|otherwise intractable problems]] in the calculation of the motions of planets in the solar system. For instance, [[Newton's law of universal gravitation]] explained the gravitation between two astronomical bodies, but when a third body is added, the problem was, "How does each body pull on each?" [[Keplerian orbit|Kepler's orbital equations]] only solve Newton's gravitational equations when the latter are limited to just two bodies interacting. The gradually increasing accuracy of [[astrometry|astronomical observations]] led to incremental demands in the accuracy of solutions to Newton's gravitational equations, which led many eminent 18th and 19th&nbsp;century mathematicians, notably [[Joseph-Louis Lagrange]] and [[Pierre-Simon Laplace]], to extend and generalize the methods of perturbation theory.

These well-developed perturbation methods were adopted and adapted to solve new problems arising during the development of [[quantum mechanics]] in 20th&nbsp;century atomic and subatomic physics. [[Paul Dirac]] developed quantum perturbation theory in 1927 to evaluate when a particle would be emitted in radioactive elements. This was later named [[Fermi's golden rule]].<ref>{{cite book |last1 = Bransden |first1 = B.H. |last2 = Joachain |first2 = C.J. |year = 1999 |title=Quantum Mechanics |edition = 2nd |isbn = 978-0-58235691-7 |page = 443 |publisher = Prentice Hall }}</ref><ref>{{cite journal |last = Dirac |first = P.A.M. |author-link = Paul Dirac |date = 1 March 1927 |title = The quantum theory of emission and absorption of radiation |journal = [[Proceedings of the Royal Society A]] |volume = 114 |issue = 767 |pages = 243–265 |doi = 10.1098/rspa.1927.0039 |doi-access = free |jstor = 94746 |bibcode = 1927RSPSA.114..243D }} See equations&nbsp;(24) and (32).</ref> Perturbation theory in quantum mechanics is fairly accessible, mainly because quantum mechanics is limited to linear wave equations, but also since the quantum mechanical notation allows expressions to be written in fairly compact form, thus making them easier to comprehend. This resulted in an explosion of applications, ranging from the [[Zeeman effect]] to the [[hyperfine splitting]] in the [[hydrogen atom]].

Despite the simpler notation, perturbation theory applied to [[quantum field theory]] still easily gets out of hand. [[Richard Feynman]] developed the celebrated [[Feynman diagram]]s by observing that many terms repeat in a regular fashion. These terms can be replaced by dots, lines, squiggles and similar marks, each standing for a term, a denominator, an integral, and so on; thus complex integrals can be written as simple diagrams, with absolutely no ambiguity as to what they mean. The one-to-one correspondence between the diagrams, and specific integrals is what gives them their power. Although originally developed for quantum field theory, it turns out the diagrammatic technique is broadly applicable to many other perturbative series (although not always worthwhile).

In the second half of the 20th&nbsp;century, as [[chaos theory]] developed, it became clear that unperturbed systems were in general [[completely integrable system]]s, while the perturbed systems were not. This promptly lead to the study of "nearly integrable systems", of which the [[KAM torus]] is the canonical example. At the same time, it was also discovered that many (rather special) [[non-linear system]]s, which were previously approachable only through perturbation theory, are in fact completely integrable. This discovery was quite dramatic, as it allowed exact solutions to be given. This, in turn, helped clarify the meaning of the perturbative series, as one could now compare the results of the series to the exact solutions.

The improved understanding of [[dynamical system]]s coming from chaos theory helped shed light on what was termed the ''small denominator problem'' or ''small divisor problem''. In the 19th&nbsp;century [[Henri Poincaré|Poincaré]] observed (as perhaps had earlier mathematicians) that sometimes 2nd and higher order terms in the perturbative series have "small denominators": That is, they have the general form <math>\ \frac{\ \psi_n V \phi_m\ }{\ (\omega_n -\omega_m)\ }\ </math> where <math>\ \psi_n\ ,</math> <math>\ V\ ,</math> and <math>\ \phi_m\ </math> are some complicated expressions pertinent to the problem to be solved, and <math>\ \omega_n\ </math> and <math>\ \omega_m\ </math> are real numbers; very often they are the [[energy]] of [[normal mode]]s. The small divisor problem arises when the difference <math>\ \omega_n - \omega_m\ </math> is small, causing the perturbative correction to "[[divergent series|blow up]]", becoming as large or maybe larger than the zeroth order term. This situation signals a breakdown of perturbation theory: It stops working at this point, and cannot be expanded or summed any further. In formal terms, the perturbative series is an ''[[asymptotic series]]'': A useful approximation for a few terms, but at some point becomes ''less'' accurate if even more terms are added. The breakthrough from chaos theory was an explanation of why this happened: The small divisors occur whenever perturbation theory is applied to a chaotic system. The one signals the presence of the other.{{citation needed|date=January 2025}}

===Beginnings in the study of planetary motion===
Since the planets are very remote from each other, and since their mass is small as compared to the mass of the Sun, the gravitational forces between the planets can be neglected, and the planetary motion is considered, to a first approximation, as taking place along Kepler's orbits, which are defined by the equations of the [[two-body problem]], the two bodies being the planet and the Sun.<ref name=EoM>{{cite web |title = Perturbation theory |website = Encyclopedia of Mathematics (encyclopediaofmath.org) |url = http://www.encyclopediaofmath.org/index.php?title=Perturbation_theory&oldid=11676 }}</ref>

Since astronomic data came to be known with much greater accuracy, it became necessary to consider how the motion of a planet around the Sun is affected by other planets. This was the origin of the [[three-body problem]]; thus, in studying the system Moon-Earth-Sun, the mass ratio between the Moon and the Earth was chosen as the "small parameter". Lagrange and Laplace were the first to advance the view that the so-called "constants" which describe the motion of a planet around the Sun gradually change: They are  "perturbed", as it were, by the motion of other planets and vary as a function of time; hence the name "perturbation theory".<ref name=EoM/>

Perturbation theory was investigated by the classical scholars – Laplace, [[Siméon Denis Poisson]], [[Carl Friedrich Gauss]] – as a result of which the computations could be performed with a very high accuracy. The [[Discovery of Neptune|discovery of the planet Neptune]] in 1848 by [[Urbain Le Verrier]], based on the deviations in motion of the planet [[Uranus]]. He sent the coordinates to [[Johann Gottfried Galle|J.G. Galle]] who successfully observed Neptune through his telescope – a triumph of perturbation theory.<ref name=EoM/>