Perturbation Theory, Introduction to

The idea behind Perturbation Theory is that when we are not able to determine exact solutions to a given problem, we might be able to determine approximate solutions to our problem starting from solutions to an approximate version of the problem, amenable to exact treatment. Thus, in a way, we use exact solutions to an approximate problem to get approximate solutions to an exact problem.

It goes without saying that many mathematical problems met in realistic situations, in particular as soon as we leave the linear framework, are not exactly solvable–either for an inherent impossibility or for our insufficient skills. Thus, Perturbation Theory is often the only way to approach realistic nonlinear systems.

It is implicit in the very nature of perturbation theory that it can only worke once a problem which is both solvable–one also says “integrable”–and in some sense “near” to the original problem can be identified (it should be mentioned in this respect that the issue of “how near...