An extended Ideal Resonance Problem

Abstract

An Extended Resonance Problem is defined by the Hamiltonian,

$$F = B(y) + 2\mu ^2 A(y)[\sin x + \lambda (y)]^2 \mu<< 1,\lambda = O(\mu ).$$

It is noted here that the phase-plane trajectories exhibit adouble libration, enclosing two centers, for the initial conditions of motion satisfying the inequality

$$1 - |\lambda |< |\alpha |< 1 + |\lambda |,$$

where α is the usualresonance parameter.

A first order solution for the case of double libration is constructed here by a generalization of the procedure previously used in solving the Ideal Resonance Problem with λ=0. The solution furnishes a reference orbit for a Perturbed Ideal Problem if a double libration occurs as a result of perturbations.