Symmetry breaking in mechanical systems

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 generators of ζ(τ,μ1) evaluated at the base points Exp(τvqe(τ,μ1)) of τvqe(τ,μ1) are relative equilibria. Here μ1 is a certain component in a direct sum decomposition of g* naturally associated to the bifurcation problem. This produces a branch of relative equilibria starting in the subspace t ·qe which has trivial isotropy for τ > 0 and which depends smoothly on the parameter μ1 ∈ g*. In the process, the precise form of the perturbation β(τ,μ) is also determined; it is a quadratic polynomial in τ whose coefficients are certain components in the direct sum decomposition of g* mentioned above. There are two technical problems in this procedure: the existence of the limit of ζ(τ,vqe,μ) as τ → 0 and the extension of the amended potential at points with symmetry. The existence of the limit is shown using the Lyapunov-Schmidt procedure. To extend the amended potential and its derivatives at points with symmetry, two auxiliary functions obtained by blow-up are introduced. The analysis breaks up in two bifurcation problems on a space orthogonal to the G-orbit. The bifurcation results of this chapter can be regarded as an extension of the thesis of Hernández [20] and the work of Hernández and Marsden [21]. The main difference is that only one single hypothesis from [21] has been retained, namely that all points given by infinitesimal generators of elements of t evaluated at qe are relative equilibria. We have eliminated an essential non-degeneracy assumption in [21], namely that G acts freely in some G-invariant neighborhood of the G-orbit of qe excluding the orbit itself. This hypothesis excludes in effect the existence of other strata near the G-orbit of qe. Our results allow the existence of different orbit types near this relative equilibrium. A second assumption that was eliminated is the hypothesis that the isotropy subgroup of qe is the circle S1. In addition, the conclusions of our final result are stronger. The main theorem of this chapter guarantees the existence of multi-parameter families of relative equilibria, whereas the aforementioned works obtain only curves of relative equilibria. The main results of this part are contained in a submitted paper. The second part of the thesis extends the results of the first part by treating the case when the principal stratum of the action has non-trivial symmetry, that is, each point on this stratum has symmetry conjugate to a non-trivial subgroup of G. The main result of this part is the existence of symmetry breaking bifurcating branches of relative equilibria with principal symmetry emanating from the set t · qe. As opposed to the situation in the first part, the amended potential criterion along the emanating branches is not applicable anymore, because each point on such a branch has nontrivial isotropy. Thus we shall use the augmented potential and the same type of techniques as in the first part to treat branches with non-trivial isotropy. However, we can obtain only bifurcating curves of relative equilibria and not multi-parameter families; we lose the explicit dependence on the momentum value along the bifurcating branch. The last part of the thesis investigates the problem of symmetry breaking in the framework of dynamical systems with symmetry on a smooth manifold. The symmetric steady state and Hopf bifurcations are considered. First, it is shown how to locally translate these symmetry breaking problems to similar ones on a vector space. This presents several advantages because of the availability of techniques specific to the linear case, such as invariant theory, where one uses the existence of a Hilbert basis of invariant polynomials for representations of compact Lie groups. Several results in the literature give then sufficient conditions for the existence of symmetry breaking bifurcations. The same method is then used in the Hamiltonian context to find sufficient conditions for the existence of symmetry breaking bifurcation. The Hamilltonian steady state (passing of eigenvalues of the linearization on the imaginary axis at the origin) and the Hamiltonian Hopf bifurcations are analyzed. This is achieved by noting that in the Hamiltonian context one can work only with invariant polynomials of the action and ignore the equivariant polynomials. In addition, it is shown how to locally translate the bifurcation problem from the manifold to the tangent space at the bifurcation point. This is done by using the equivariant Darboux theorem.