All relative equilibria of Hamiltonian in detuned 1:2:3 resonance

Abstract

This paper deals with three degrees of freedom Hamiltonian system in the case that the frequencies of the linearization are in 1:2:3 resonance. We obtain the truncated second order normal form expressed with the basic invariants. Also, we identify the coefficients of the terms that remain in normalization procedure. We concentrate on determining the relative equilibria with or without considering a perturbation in frequencies. By a homogeneous structure, all relative equilibria are computed and the linear stability of isolated relative equilibria is investigated. Especially we find some parametric manifold of relative equilibria and a quintic polynomial which determines relative equilibria.