Perturbation theory tackles hard problems by nudging easy ones, one correction at a time
The Moon's orbit defeated even Isaac Newton, who reportedly complained that the problem made his head ache. Earth and Sun both tug on it, so no tidy formula fits. The fix mathematicians devised was to solve a simpler version exactly, then add small corrections, an approach now called perturbation theory.
The method starts by splitting a problem into two parts: one that can be solved exactly and a small leftover, the perturbation. The answer is then written as a series in a small parameter, often labelled epsilon. The first term is the known solution, and each later term, multiplied by a higher power of epsilon, adds a finer correction. In practice people often stop after two terms, the exact solution plus a first-order adjustment, since the rest are usually tiny.
Celestial mechanics was its first home. Under Newton's gravity, a Keplerian ellipse describes two bodies perfectly, say Earth and Moon, but not three or more, and not once general relativity is taken into account. Astronomers therefore began with the ellipse and worked out the Sun's pull as a correction. The calculations are mechanical but quickly sprawl into a dizzying number of terms, and much of the craft lies in taming them.
Not every series behaves. When it converges, the problem is called regular and the approximation slides smoothly towards the true answer. Some series diverge yet still give excellent results if cut off at the point where their terms are smallest; these are asymptotic series. Problems whose expansions diverge or need fractional powers are called singular and demand special techniques.
The approach now runs through physics. Common starting points include harmonic oscillators, linear waves and systems of particles that do not interact, with perturbations adding the messy nonlinear or interacting parts. It computes particle paths, magnetisation and the ground-state energy of quantum systems. Its most elaborate form lives in quantum field theory, where Feynman diagrams turn each term of a perturbation series into a picture that physicists can manipulate.
Source: Perturbation theory
