Classical dynamics of the 1:1, 1:2 and 1:3 resonance Hamiltonians Original Research Article

In this paper, the classical dynamics of the resonance Hamiltonian E = ω1I1 + ω2I2 + x11I12 + x22I22 + x12I1I2 + 2kmnI1m/2I2n/2 cos(mϕ1 − nϕ2), where m = 1 and n = 1, 2 and 3 is thoroughly investigated. Real expressions are obtained for the solutions of Hamiltonʹs equations, for the fundamental frequencies of the tori supporting the trajectories and for the action integrals of the system. It is shown that the number of possible phase space structures, obtained by varying the energy while keeping I (the classical equivalent of the quantum polyad number) constant is limited to 2 for the 1:1 and the 1:3 resonances and to 3 for the 1:2 resonance, and that each structure containes at most one hyperbolic fixed point separatrix. Asymptotic expressions for the fundamental frequencies in the neighborhood of the hyperbolic point are given. The calculation of action integrals presented previously is discussed.