# Generally applicable development of classical perturbation theory

1. Jan 13, 2014

### HJ Farnsworth

Greetings,

Does anyone know of some good sources that explain classical perturbation theory, preferably using the Lagrangian formalism?

The sources that I have seen more-or-less say, "write $L=L_{0}+λδL$, where $L_{0}$ is an unperturbed, soluble Lagrangian, $δL$ is the perturbation, and $λ$ is a small physical parameter. Then, find the new solution as $q(t)=q_{0}(t) +λq_{1}(t)+λ^{2}q_{2}(t)+...$". Examples are usually given, but it is not clear to me what the general scheme of the perturbation theory is. The way $λ$ is chosen usually seems arbitrary to me, and the fact that it is a physical, rather than mathematical, parameter concerns me, since this usually means that it is really just some small finite value, so that even the infinitely-expanded $q(t)$ does not truly represent an exact solution. Furthermore, unlike perturbation theory in quantum mechanics, I cannot find anything like a general recursive technique (and the derivation of that recursive technique) to find higher-order corrections from lower-order corrections, or anything like that. Perhaps there is no such general method in classical mechanics, but it seems like there must be something more developed than what I have read.

So basically, the developments of classical perturbation theory that I have seen are developed no further than the core concept of perturbation theory itself (use a small parameter and an expansion). In specific examples, the approach used for that example alone is developed, but I haven't seen anything general that can be applied to a wide class of problems.

Does anyone know of anything like this (a full, relatively general, preferably recursive development of classical perturbation theory, ideally using the Lagrangian formalism), and if so, could you please explain it a bit, or point me to a source that explains it?

Thank you very much for any help that you can give.

-HJ Farnsworth