Lagrangian fibrations in coupled resonant oscillators in the paired case of four degrees of freedom containing swinging spring oscillator

Abstract

This paper deals with Lagrangian fibrations in coupled resonant oscillators containing swinging spring oscillators of four degrees of freedom. We will consider the cases with zero angular momenta with 1 : 1 and 1 : 2 (Fermi) resonances. Moreover, we also consider a coupled spatial swinging spring oscillator with a harmonic oscillator. Actually, they make different four degrees of freedom Hamiltonian resonances, i.e. 1 : 2 : 1 : 1 and 1 : 2 : 1 : 2 resonances. We will obtain the third-order normalization of these Hamiltonian resonances. We consider the linear Casimir energy and we approximate models by normal forms. Then our models are Liouville integrable. For one case with zero angular momentum and coupled swinging spring, we will identify Hopf fibration. The relative equilibria and normal modes are identified in the space of orbits. Our analysis is based on affecting energy levels in oscillators. These energy levels have been marked with distinguished parameters which show the constants of motions or integrals.