### perturbation theory

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

**Perturbation theory** is a generalization of
a mechanism for orbiting three body problems,
by breaking it into two simple two-body orbiting problems
along with an unsolvable but approximatable **perturbations**
(slight changes) of the two orbits,
i.e., how each orbit is slightly modified over time
by the presence of the other body.
As a theory, it has been generalized
to handle other unsolvable equations (often differential
equations) of other kinds of physical systems,
in chemistry, quantum mechanics, etc.
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 estimations
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*)
http://en.wikipedia.org/wiki/Perturbation_theory

**Referenced by:**

Laplace-Lagrange secular theory

Lie transform

N-body problem

Zel'dovich approximation

index