Rigorous estimates for the series expansions of Hamiltonian perturbation theory

In the present paper we prove a theorem giving rigorous estimates in the problem of bringing to normal form a nearly integrable Hamiltonian system, using methods of classical perturbation theory, i.e. series expansions in the "small parameter" ε. For any order of normalization, we give a lower bound e{open}*r for the convergence radius of the normalized Hamiltonian, and a greater bound for the remainder, i.e. the non normalized part of the Hamiltonian. As an application, we consider the case of weakly coupled harmonic oscillators with highly nonresonant frequencies and show how, by optimizing, for fixed ε, the order r of normalization, one gets for the remainder a greater bound of the form Ae-(e{open}*1/e{open})a, with positive constants A, a and e{open}1* exponential estimate of Nekhoroshev's type. © 1985 D. Reidel Publishing Company.