Analytical treatment of a non-integrable pendulum based on eigenvalue problem of the Liouville operator

We perform analytical and quantitative analyses of the motion of a non-integrable pendulum with two degrees of freedom, in which an integrable nonlinear pendulum and a harmonic oscillator are weakly coupled through a non-integrable perturbative interaction, based on the eigenvalue problem of the Liouvillian, which is the generator of time evolution in classical mechanics. The eigenfunctions belonging to the zero eigenvalue of the Liouvillian correspond to the invariants of the motion. The zero eigenvalue of the integrable unperturbed Liouvillian is infinitely degenerate at the resonance point. By applying a perturbation, level repulsion occurs between the eigenstates of the unperturbed system, and some of the degeneracy is lifted, resulting in a non-zero eigenvalue. In order to evaluate the frequency gap caused by the level repulsion, we introduce an auxiliary operator called the collision operator, which is well known in non-equilibrium statistical mechanics. We show that the dependence of the magnitude of a frequency gap on the coupling constant can be quantitatively evaluated by simply finding the condition for the existence of the collision operator, without directly solving the eigenvalue problem.