### perturbation theory

(breaking an equation into a solvable part and approximable part)

**Perturbation theory** is a generalization of
a mechanism used for orbiting three-body problems,
by breaking it into two simple two-body orbiting problems
along with analytically-unsolvable but approximable **perturbations**
(slight changes) of the two orbits,
i.e., how each orbit is slightly modified by the presence of the other body,
and the end result of these tiny-but-repeated modifications.
As a theory, it has been generalized
to handle other unsolvable equations (often, differential
equations) for other kinds of physical systems,
including chemistry, quantum mechanics, and numerous other problem areas.
The perturbation is cast as a series of ever-smaller
solvable equations, e.g., like a Taylor series.

A **first order perturbation problem**
(or **regular perturbation problem**)
is a problem that includes a very small parameter and
the solution can be found by approximating that parameter as zero.
If such an approximation is too far off,
it is known as a **second order perturbation problem**
(or **singular perturbation problem** or **degenerate perturbation problem**).

The Hamiltonian is useful in perturbation theory to study secular
(long term) motions related to planetary orbits, and is useful
in series-expanded form to provide tractable estimates
and because it can handle coordinate transformations that simplify
solutions, sometimes effectively eliminating a coordinate. With a
multipole expansion, terms through the **octupole term**
can be necessary to explain observed exotic extra-solar planet orbits.

(*orbits,dynamics,mathematics*)
**Further reading:**

https://en.wikipedia.org/wiki/Perturbation_theory

**Referenced by pages:**

celestial mechanics

Keplerian orbit

Laplace-Lagrange secular theory

Lie transform

minimum orbit intersection distance (MOID)

N-body problem

quantum field theory (QFT)

Zel'dovich approximation

Index