Symmetry and resonance in Hamiltonian systems

Abstract

In this paper we study resonances in two degrees of freedom, autonomous, hamiltonian systems. Due to the presence of a symmetry condition on one of the degrees of freedom, we show that some of the resonances vanish as lower order resonances. After determining the size of the resonance domain, we investigate this order change of resonance in a rather general potential problem with discrete symmetry and consider as an example the Henon-Heiles family of hamiltonians. We also study a classical example of a mechanical system with symmetry, the elastic pendulum, which leads to a natural hierarchy of resonances with the 4 : 1-resonance as the most prominent after the 2 : 1-resonance and which explains why the 3 : 1-resonance is neglected.