Relative normal modes for nonlinear Hamiltonian systems

Abstract

An estimate on the number of distinct relative periodic orbits around a stable relative equilibrium in a Hamiltonian system with continuous symmetry is given. This result constitutes a generalization to the Hamiltonian symmetric framework of a classical result by Weinstein and Moser on the existence of periodic orbits in the energy levels surrounding a stable equilibrium. The estimate obtained is very precise in the sense that it provides a lower bound for the number of relative periodic orbits at each prescribed energy and momentum values neighbouring the stable relative equilibrium in question and with any prefixed (spatio-temporal) isotropy subgroup. Moreover, it is easily computable in particular examples. It is interesting to see how, in our result, the existence of non-trivial relative periodic orbits requires (generic) conditions on the higher-order terms of the Taylor expansion of the Hamiltonian function, in contrast with the purely quadratic requirements of the Weinstein–Moser theorem, which emphasizes the highly nonlinear character of the relatively periodic dynamical objects.