[1][2] A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts.
[3] In regular perturbation theory, the solution is expressed as a power series in a small parameter
represent the first-order, second-order, third-order, and higher-order terms, which may be found iteratively by a mechanistic but increasingly difficult procedure.
[1][2] The earliest use of what would now be called perturbation theory was to deal with the otherwise unsolvable mathematical problems of celestial mechanics: for example the orbit of the Moon, which moves noticeably differently from a simple Keplerian ellipse because of the competing gravitation of the Earth and the Sun.
[4] Perturbation methods start with a simplified form of the original problem, which is simple enough to be solved exactly.
The perturbative expansion is created by adding successive corrections to the simplified problem.
The corrections are obtained by forcing consistency between the unperturbed solution, and the equations describing the system in full.
In practice, this process rapidly explodes into a profusion of terms, which become extremely hard to manage by hand.
Isaac Newton is reported to have said, regarding the problem of the Moon's orbit, that "It causeth my head to ache.
"[5] This unmanageability has forced perturbation theory to develop into a high art of managing and writing out these higher order terms.
Perturbation theory has been used in a large number of different settings in physics and applied mathematics.
Examples of the kinds of solutions that are found perturbatively include the solution of the equation of motion (e.g., the trajectory of a particle), the statistical average of some physical quantity (e.g., average magnetization), and the ground state energy of a quantum mechanical problem.
Examples of exactly solvable problems that can be used as starting points include linear equations, including linear equations of motion (harmonic oscillator, linear wave equation), statistical or quantum-mechanical systems of non-interacting particles (or in general, Hamiltonians or free energies containing only terms quadratic in all degrees of freedom).
For physical problems involving interactions between particles, the terms of the perturbation series may be displayed (and manipulated) using Feynman diagrams.
Perturbation theory was first devised to solve otherwise intractable problems in the calculation of the motions of planets in the solar system.
The gradually increasing accuracy of astronomical observations led to incremental demands in the accuracy of solutions to Newton's gravitational equations, which led many eminent 18th and 19th 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 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 resulted in an explosion of applications, ranging from the Zeeman effect to the hyperfine splitting in the hydrogen atom.
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).
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 systems, which were previously approachable only through perturbation theory, are in fact completely integrable.
In the 19th century 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
is small, causing the perturbative correction to "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.
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".
[12] 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.
Calculations to second, third or fourth order are very common and the code is included in most ab initio quantum chemistry programs.
A shell-crossing (sc) occurs in perturbation theory when matter trajectories intersect, forming a singularity.