An extended Ideal Resonance Problem

An Extended Resonance Problem is defined by the Hamiltonian, $$F = B(y) + 2mu ^2 A(y)[sin x + lambda (y)]^2 mu<< 1,lambda = O(mu ).$$ 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.