Métamorphoses de systèmes Hamiltoniens avec symétries


In this work we study four classical Hamiltonian systems with discrete or continuous symmetries using methods and techniques that have been developed in the last decades. Three of these systems are Hamiltonian families which model molecular and atomic systems. These systems are the triply degenerate vibrational mode of tetrahedral molecules, the hydrogen atom in crossed electric and magnetic fields and the floppy molecule LiCN. The hydrogen atom is described naturally as a two parameter family where the parameters are the strengths of the two fields. The other two physical systems are described as specific members of more general parametric families. We use normalization (when appropriate) and reduction in order to reduce the number of degrees of freedom of these families. We focus on certain qualitative characteristics of these systems, namely, relative equilibria, Hamiltonian Hopf bifurcations and monodromy and the metamorphoses of these characteristics in different parameter regions. The fourth system is a perturbation of two harmonic oscillators in $1:-2$ resonance. Such a system may describe the dynamics in the neighbourhood of an equilibrium of a two degree of freedom Hamiltonian. For this system we give an analytic proof of the existence of fractional monodromy, which is a radical generalization of standard monodromy.