Invariants of Nearly Periodic Hamiltonian Systems

A new and simple method of finding an invariant J of a nearly periodic dynamical system is presented. The Hamiltonian is written as H = p1 + εΩ(qipi), where Ω is periodic in q1 and ε « 1. The first four terms of the invariant series are found explicitly in terms of Ω using Poisson bracket and averaging operators. This invariant is related to the adiabatic invariant and to various constants of motion discussed in celestial mechanics, such as Whittaker's adelphic integral. J is shown to be an asymptotic constant by using the rigorous methods of Kruskal to calculate the adiabatic invariant K; it is found that K/τ = H - εJ, where τ is the period in q1. The adelphic integral has different functional forms depending on the presence of resonant denominators, but is shown to be always a function of H and J. The present method provides a single functional form which is applicable even when Ω is only almost periodic in q1. It is also much simpler than the methods of adiabatic invariant theory.