Averaging method and adiabatic invariants

There are many problems that lead to analysis of Hamiltonian dynamical systems in which one can distinguish motions of two types: slow motions and fast motions. Adiabatic perturbation theory is a mathematical tool for the asymptotic description of dynamics in such systems. This theory allows to construct adiabatic invariants, which are approximate first integrals of the systems. These quantities change by small amounts on large time intervals, over which the variation of slow variables is not small. Adiabatic invariants usually arise as first integrals of the system after having been averaged over the fast dynamics. Adiabatic invariants are important dynamical quantities. In particular, if a system has sufficiently many adiabatic invariants. then the motion over lone time intervals is close to regular. On the other hand, the destruction of adiabatic invariance leads to chaotic dynamics.