The perturbed ideal resonance problem

Abstract

The author's previous studies concerning the Ideal Resonance Problem are enlarged upon in this article. The one-degree-of-freedom Hamiltonian system investigated here has the form

$$\begin{array}{*{20}c} { - F = B(x) + 2\mu ^2 A(x)\sin ^2 y + \mu ^2 f(x,y),} \\ {\dot x = - F_y ,\dot y = F_x .} \\ \end{array}$$

The canonically conjugate variablesx andy are respectively the momentum and the coordinate, andμ ² is a small positive constant parameter. The perturbationf is o (A) and is represented by a Fourier series iny. The vanishing of ∂B/∂x≡B ⁽¹⁾ atx=x ₀ characterizes the resonant nature of the problem.

With a suitable choice of variables, it is shown how a formal solution to this perturbed form of the Ideal Resonance Problem can be constructed, using the method of ‘parallel’ perturbations. Explicit formulae forx andy are obtained, as functions of time, which include the complete first-order contributions from the perturbing functionf. The solution is restricted to the region of deep resonance, but those motions in the neighbourhood of the separatrix are excluded.