Publication
The first part of this thesis studies the problem of symmetry breaking in the context of simple mechanical systems with compact symmetry Lie group G. In this part we shall assume that the principal stratum of the G-action on the configuration space Q of a given mechanical system is associated to the trivial subgroup {e}. Let T be a maximal torus of G whose Lie algebra is denoted by t. Let qe ∈ Q be a given point with nontrivial symmetry Gqe ≠ {e} and assume that Gqe ⊆ T. We shall make the hypothesis that the values of the infinitesimal generators of elements in t at qe are all relative equilibria of the given mechanical system. These relative equilibria form a vector subspace t · qe of TqeQ. As will be shown, every relative equilibrium in this subspace has symmetry equal to Gqe . The goal of this part of the thesis is to give sufficient conditions capable to insure the existence of points in this subspace from which symmetry breaking branches of relative equilibria with trivial symmetry will emerge. Sufficient Lyapunov stability conditions along these branches will be given if G = T. The strategy of the method can be roughly described as follows. Take a regular element μ ∈ g* which happens to be the momentum value of some relative equilibrium defined by an element of t. Choose a one parameter perturbation β(τ, μ) ∈ g* of μ that lies in the set of regular points of g*, for small values of the parameter τ > 0. Consider the Gqe-representation on the tangent space TqeQ. Let vqe be an element in the principal stratum of the representation and also in the normal space to the tangent space at qe to the orbit G · qe. Assume that its norm is small enough in order for vqe to lie in the open ball centered at the origin 0qe ∈ TqeQ where the Riemannian exponential is a diffeomorphism. The curve τvqe projects by the exponential map to a curve qe(τ) in a neighborhood of qe in Q whose value at τ = 0 is qe. We search for relative equilibria in TQ starting at points in t · qe such that their base curves in Q equal qe(τ) and their momentum values are β(τ, μ). Not all possible perturbations β(τ,μ) are allowed and it is part of the problem to determine which ones will yield symmetry breaking bifurcating branches of relative equilibria. To do this, let ζ(τ,vqe,μ) ∈ g be the image of β(τ,μ) by the inverse of the locked inertial tensor of the mechanical problem under consideration evaluated at qe(τ) for τ > 0. If one can show that the limit ζ(0, vqe,μ) of ζ(τ,vqe,μ) exists and belongs to t for τ → 0, then the infinitesimal generator of ζ(0,vqe,μ) evaluated at qe is automatically a relative equilibrium since it belongs to t · qe. We shall determine an open Gqe-invariant neighborhood U of the origin in the orthogonal complement to the tangent space to the orbit G · qe such that this limit exists whenever vqe ∈ U. Next, we will determine a family vqe(τ,μ1) ∈ TQ and, among all possible ζ(τ, vqe,μ), another family ζ(τ,μ1) ∈ g such that the infinitesimal generat