Approximate constants of motion for classically chaotic vibrational dynamics: Vague tori, semiclassical quantization, and classical intramolecular energy flow

Abstract

Coupled nonlinear Hamiltonian systems are known to exhibit regular (quasiperiodic) and chaotic classical motions. In this and the preceding paper by Jaffé and Reinhardt, we find substantial short time regularity even in the chaotic regions of phase space for what appears to be a large class of systems. This regularity is demonstrated by the behavior of approximate constants of motion calculated by Padé summation of the Birkhoff–Gustavson normal form expansion and is attributed to remnants of destroyed invariant tori in phase space. The remnant toruslike manifold structures are used to suggest justification for use of Einstein–Brillouin–Keller semiclassical quantization procedures for obtaining quantum energy levels even in the absence of complete tori and to form a theoretical basis for the calculation of rate constants for intramolecular mode–mode energy transfer. These results are illustrated in a thorough analysis of the Hénon–Heiles oscillator problem. Possible generality of the analysis is shown by brief consideration of classical dynamics for the Barbanis Hamiltonian, Zeeman effect in hydrogen, and recent results of Wolf and Hase for the H–C–C fragment.