IMBEDDINGS OF INTEGRAL SUBMANIFOLDS AND ASSOCIATED ADIABATIC INVARIANTS OF SLOWLY PERTURBED INTEGRABLE HAMILTONIAN SYSTEMS

A new method is developed for characterizing the evolution of invariant tori of slowly varying perturbations of completely integrable (in the sense of LiouvilleArnold [1-3]) Hamiltonian systems on cotangent phase spaces. The method is based on Cartanʹs theory of integral submanifolds, and it provides an algebro-analytic approach to the investigation of the embedding [4-10] of the invariant tori in phase space that can be used to describe the structure of quasi-periodic solutions on the tori. In addition, it leads to an adiabatic perturbation theory [3,12,13] of the corresponding Lagrangian asymptotic submanifolds via the Poincare-Cartan approach [4], a new Poincare-Melnikov type [5,11,14] procedure for determining stability, and fresh insights into the existence problem for adiabatic invariants [2,3] of the Hamiltonian systems under consideration.