We present a detailed analysis of the orbital stability of the Pais-Uhlenbeck oscillator, using Lie-Deprit series and Hamiltonian normal form theories. In particular, we explicitly describe the reduced phase space for this Hamiltonian system and give a proof for the existence of stable orbits for a certain class of self-interaction, found numerically in previous works, by using singular symplectic reduction.
Abstract. We study the determination of the second-order normal form for perturbed Hamiltonians2 H 2 , relative to the periodic flow of the unperturbed Hamiltonian H 0 . The formalism presented here is global, and can be easily implemented in any CAS. We illustrate it by means of two examples: the Hénon-Heiles and the elastic pendulum Hamiltonians.
We present a coordinate-free approach for constructing approximate first integrals of generalized slow-fast Hamiltonian systems, based on the global averaging method on parameter-dependent phase spaces with S 1 −symmetry. Explicit global formulas for approximate second-order first integrals are derived. As examples, we analyze the case quadratic in the fast variables (in particular, the elastic pendulum), and the charged particle in a slowly-varying magnetic field.
In the context of normal forms, we study a class of slow–fast Hamiltonian systems on general Poisson fiber bundles with symmetry. Our geometric approach is motivated by a link between the deformation theory for Poisson structures on fibered manifolds and the adiabatic perturbation theory. We present some normalization results which are based on the averaging theorem for horizontal 2-cocycles on Poisson fiber bundles.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.