Classical Hamiltonian perturbation theory without secular terms or small denominators

Abstract

We consider perturbation theory in ϵ for the classical Hamiltonian H = H 0 + ϵH 1, where H 0 gives rise to a known motion and ϵ is small. First we demonstrate how the usual secular terms and small denominators arise from a straightforward expansion in ϵ and argue that they are artifacts of the method. Then we present an alternative perturbation theory based on an analysis of the operator ( s − L) −1, where s is a complex number and L is the Liouville operator corresponding to H. This perturbation series contains neither secular terms nor small denominators. In the case of almost multiply periodic systems we show, to lowest non-trivial order in ϵ, how our series reproduces the standard results both in the resonant and nonresonant regions — all in one analytic formula. As a final exercise we demonstrate that energy is conserved at order ϵ n+1 when the accuracy of the theory is order ϵ n .