Perturbation theory in classical mechanics We are familiar with perturbation theory in classical mechanics in the form of canonical perturbation theory and Lie transform theory. However, is there a perturbation theory within the Lagrangian framework? If so, can we briefly summarise how it looks; if there isn't, why not?
 A: It depends what kind of perturbation theory you're talking about.
There is a simple "equation-of-motion" perturbation theory, which starts with the equations of motion and then solves iteratively.  As the EoMs can usually be obtained using a Lagrangian, you have a bona fide Lagrangian perturbation theory.   A well-known technique is the Poincare-Lindstedt method.
For instance the solution to the EoM of a perturbed oscillator
\begin{align}
\ddot{x}+x+\epsilon x^3
\end{align}
is solved by expanding $x$ in powers of $\epsilon$ and expanding the frequency of oscillation $\omega$  in powers of $\epsilon$:
\begin{align}
x(t)&=x_0(t)+\epsilon x_1(t)+\ldots\, ,\\
\omega&=\omega_0+\epsilon \omega_1(t)+\ldots.
\end{align}
The expansion of $\omega$ is done to avoid so-called secular terms that appear (more or less systematically) in more direct (Fourier-based) approach.
The more canonical formalism is based on Hamilton-Jacobi theory, and uses action-angle variables as its starting point.  Again it is based on an expansion but this time the perturbative series in $\epsilon$ seeks to find a sequence of canonical transformations that will eliminate the new angle variable
\begin{align}
H(I,\theta)&\to K(J;\epsilon)\, ,\\
K(J;\epsilon)&=K_0(J)+\epsilon K_1(J)+\ldots
\end{align}
with an infinitesimal canonical transformation
\begin{align}
F_2(J,\phi)&=J\theta+\epsilon W^{(1)}(J,\theta)+\ldots\, ,\\
\phi&=\frac{\partial F_2}{\partial J}=\theta+\epsilon \frac{\partial W^{(1)}(J,\theta)}{\partial J}+\ldots
\end{align}
Sir Michael Berry has a good set of lecture notes on this in

Berry, Michael Victor. "Regular and irregular motion." AIP Conference proceedings. Vol. 46. No. 1. American Institute of Physics, 1978.

and available from his website.
