Averaging for Hamiltonian Systems with One Fast Phase and Small Amplitudes

In this paper we consider an analytic Hamiltonian system differing from an integrable system by a small perturbation of order $\varepsilon$. The corresponding unperturbed integrable system is degenerate with proper and limit degeneracy: all variables, except two, are at rest and there is an elliptic singular point in the plane of these two variables. It is shown that by an analytic symplectic change of the variable, which is $O(\varepsilon)$-close to the identity substitution, the Hamiltonian can be reduced to a form differing only by exponentially small ($O(e^{-\operatorname{const}/\varepsilon})$) terms from the Hamiltonian possessing the following properties: all variables, except two, change slowly at a rate of order $\varepsilon$ and for the two remaining variables the origin is the point of equilibrium; moreover, the Hamiltonian depends only on the action of the system linearized about this equilibrium.