Canonical Perturbation Theory with Several Degrees of Freedom

Abstract

We extend the perturbation theory of the previous chapter by going one order further and permitting several degrees of freedom. So let the unperturbed problem $H_{0}(J_{k}^{0})$ be solved. Then we expand the perturbed Hamiltonian in the $(w_{k}^{0},J_{k}^{0})$-“basis” according to

$$\displaystyle{ H{\bigl (w_{k}^{0},J_{ k}^{0}\bigr )} = H_{ 0}{\bigl (J_{k}^{0}\bigr )} +\varepsilon H_{ 1}{\bigl (w_{k}^{0},J_{ k}^{0}\bigr )} +\varepsilon ^{2}H_{ 2}{\bigl (w_{k}^{0},J_{ k}^{0}\bigr )} +\,\ldots \;. }$$

(11.1)

We are looking for the generating function of the canonical transformation which will lead us from the variables $(J_{k}^{0},w_{k}^{0})$ to the new variables $(J_{k},w_{k})$.