Global Results by Local Averaging for Nearly Hamiltonian Systems

Abstract

This chapter presents the global results by local averaging for nearly Hamiltonian systems. It presents a space in which the points represent differential equations; a subset consists of Hamiltonian systems, and a further subset of integrable Hamiltonian systems. Integrable systems do not form an open set, either among Hamiltonian systems or in general, and it is natural to ask what sort of systems lay near an integrable system. For the Hamiltonian case, this leads into the theories of Poincarè–Birkhoff and Kolmogorov–Arnol’d–Moser. The chapter presents a case of non-Hamiltonian perturbations of an integrable Hamiltonian system. According to a general theorem of Arnol’d, the phase space of an integrable system is divided up by separatrices into regions admitting action/angle variables r, θ in which the system takes the form r = 0, θ = Ω(r). Each torus r = const is invariant and foliated with minimal invariant tori of lower dimension depending on the commensurability relations satisfied by Ω(r).