Time-Independent Canonical Perturbation Theory

Abstract

First we consider the perturbation calculation only to first order, limiting ourselves to only one degree of freedom. Furthermore, the system is to be conservative, ∂H/∂t = 0, and periodic in both the unperturbed and perturbed case. In addition to periodicity, we shall require the Hamilton-Jacobi equation to be separable for the unperturbed situation. The unperturbed problem H₀(J₀) which is described by the action-angle variables J₀ and w₀ will be assumed to be solved. Thus we have, for the unperturbed frequency:

$$ {v_0} = \frac{{\partial {H_0}}}{{\partial {J_0}}} $$

(8.1)

and

$$ {w_0} = {v_0}t + {\beta _0}\;{\rm{.}} $$

(8.2)