New approach to understanding quasiperiodicity in nonintegrable Hamiltonian systems

A classical equivalent of the Born-Oppenheimer approximation, commonly used in molecular quantum mechanics, is demonstrated to supply highly visualizable dynamic potentials which make obvious the localization of regular trajectories in nonintegrable Hamiltonian systems. Simultaneously, the approximation supplies a zeroth-order Hamiltonian H₀ with a small correction H₁ which is implicitly called for in perturbation analysis that assigns local actions. The smallness of H₁ can be used without extensive calculation to study the regions of action space where regularity exists. The methods used supply a unity to classical and quantum treatments of regularity in Hamiltonian systems. The basic ideas of the presented approach are illustrated on the example of librators and precessors of the H\'enon-Heiles system.