Stochasticity of Smooth Dynamical Systems. The Elements of KAM-Theory

Abstract

A Hamiltonian system with n degrees of freedom and with a Hamiltonian H(p ₁ , ..., p _n , q ₁ , ..., q _n ) is called integrable if it has n first integrals I ₁ = H, I ₂..., I _n which are in involution. The famous Liouville theorem states (see [Arl], [KSF]) that if the n-dimensional manifold obtained by fixing the values of the integrals I ₁ = C ₁ I ₂ = C ₂ , ...,I _n = C _n is compact, and these integrals are functionally independent in some neighbourhood of the point (C ₁ , C _n ) then this manifold is the n-dimensional torus. One could introduce cyclic coordinates φ ₁ , φ _n such that the equations of motion would obtain a simple form φ ₁ = F _i (I ₁ , ..., I _n ) = const, 1 ≤ i ≤ n, and the motion would be quasi-periodic with n frequencies.