Nearly integrable system

In this way, the above statement may be acceptable not only in a qualitative level, but the most important implication of the mathematical statement is that the character of freezing has been shown to be quantitatively the same as adiabaticity predicted by the Nekhoroshev estimate in the nearly integrable system. This solves, at least partially, question (iii) posed in the previous section, because the model Hamiltonian [Eq. (1)] just describes the situation where not all the degrees of freedom do not necessarily show adiabaticity but only a limited number of variables, just the energies of two subsystems in this case, are almost frozen. It is true that there may be, in principle, many other possibilities and the proposed one is not a unique way as for the division of phase space into lower-dimensional subspaces, but the separation induced by the internal structure of molecules is the most natural and plausible candidate. 

Theorem[Nekhoroshev]. Consider nearly integrable systems with n degrees of freedom ... 

The hierarchy of tori is theoretically predicted by the Poincare-Birkhoff theorem [10] in nearly integrable systems with two degrees of freedom. For instance, the hierarchy in the Henon-Heiles system represented by the Hamiltonian is shown in Fig. 1. 

CANONICAL PERTURBATION THEORY FOR NEARLY INTEGRABLE SYSTEMS... 

Introduction to the canonical perturbation theory for nearly integrable systems... 

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