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 H ₀(J ₀) which is described by the action-angle variables J ₀ and w ₀ will be assumed to be solved. Thus we have, for the unperturbed frequency:

$$\displaystyle{ \nu _{0} = \frac{\partial H_{0}} {\partial J_{0}} }$$

(10.1)

and

$$\displaystyle{ w_{0} =\nu _{0}t +\beta _{0}\;. }$$

(10.2)

Then the new Hamiltonian reads, up to a perturbation term of first order:

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

(10.3)

where $\varepsilon$ is a small parameter.