Estimates for non-resonant normal forms in Hamiltonian perturbation theory

We make a remark about an estimate of the rest for the non-resonant action-angle normal forms and exhibit a simple example suggesting the optimally of this estimate when there are no small divisors. Given a polynomial perturbation ol degree P and an integer k, calling y the size of the small denominators up to order k, we prove that the kth order remainder is bounded by (2/ε0)k+1 with (ε0 = Const y2/(kP2). Thus, fixing the degree of the perturbation, if y is independent of k (i.e., if there are no small divisors), we obtain a rest bounded by (const k)k+1. These estimates are also applied to the case in which the small divisors are absent, and they are conjectured to be optimal in this context. To support this idea we present a simplified model problem with no small denominators, formally related to the above calculations, and we show that it indeed has factorial divergence of its Birkholf series. We also obtain Nekhoroshev's Theorem for harmonic oscillators. We hope that our simple approach makes more accessible to a general audience this important (although quite technical) topic.