Generally applicable development of classical perturbation theory

[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

 

[eljose79]

Dear HJ Farnsworth,

Thank you for your inquiry regarding classical perturbation theory and the Lagrangian formalism. As a fellow scientist, I can understand your concerns and frustrations with the current sources available on this topic. I have personally encountered similar issues in my own research, and after some extensive searching, I have found a few sources that may be of interest to you.

Firstly, I recommend checking out the book "Classical Mechanics" by Herbert Goldstein. This book provides a comprehensive and rigorous treatment of classical mechanics, including perturbation theory and its applications. It covers both the Lagrangian and Hamiltonian formalisms and provides a detailed explanation of perturbation theory using both approaches. It also includes several examples and exercises to help solidify the concepts. I believe this book will provide you with a more thorough understanding of perturbation theory in classical mechanics.

Another source that may be helpful is the paper "Perturbation Theory in Classical Mechanics" by M. B. Green and J. H. Schwarz. This paper provides a more mathematical treatment of perturbation theory, including a discussion on the choice of the perturbation parameter and the derivation of the higher-order corrections. It also includes a section on the Lagrangian formalism and its application to perturbation theory. I believe this paper will provide you with a more in-depth understanding of the mathematical foundations of perturbation theory in classical mechanics.

Lastly, I suggest looking into the work of Pierre-Simon Laplace and his "Method of Variation of Parameters". This method provides a general approach to solving perturbed equations of motion, and it can be applied to a wide range of problems in classical mechanics. Although it may not be a fully recursive technique, it does provide a systematic way of finding higher-order corrections from lower-order corrections. You can find more information on this method in the book "A Treatise on Celestial Mechanics" by Laplace.

I hope these sources will be helpful to you in your research. If you have any further questions or need clarification on any of the concepts, please do not hesitate to reach out.