Y. Prykarpatsky et al., Imbeddings of integral submanifolds and associated adiabatic invariants ofslowly perturbed integrable Hamiltonian systems, REP MATH PH, 44(1-2), 1999, pp. 171-182
A new method is developed for characterizing the evolution of invariant tor
i of slowly varying perturbations of completely integrable (in the sense of
Liouville-Arnold [1-3]) Hamiltonian systems on cotangent phase spaces. The
method is based on Cartan's theory of integral submanifolds, and it provid
es 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 str
ucture of quasi-periodic solutions on the tori. In addition, it leads to an
adiabatic perturbation theory [3,12,13] of the corresponding Lagrangian as
ymptotic submanifolds via the Poincare-Cartan approach [4], a new Poincare-
Melnikov type [5,11,14] procedure for determining stability, and fresh insi
ghts into the existence problem for adiabatic invariants [2,3] of the Hamil
tonian systems under consideration.