Instability of Equilibrium Solutions of Hamiltonian Systems with n-Degrees of Freedom Under the Existence of Multiple Resonances and an Application to the Spatial Satellite Problem

Abstract

In this paper, we prove the instability of one equilibrium point in a Hamiltonian system with n-degrees of freedom under two assumptions: the first is the existence of multiple resonance of odd order s (without resonance of lower order) but with the possible existence of resonance of higher order; and the second is the existence of an invariant ray solution of the truncated Hamiltonian system up to order s. It is shown that in the case of resonance without interaction, the necessary conditions for instability have important simplifications with respect to the general case. Examples in three, four and six degrees of freedom are given. An application of our main result to the spatial satellite problem is considered.