Bifurcation and stability of relative equilibria with isotropy in Lagrangian systems with symmetry.

摘要

This thesis is a study of symmetric relative equilibria in Lagrangian systems with Lie group symmetry. The material covered can be grouped into three parts.; The first part is concerned with stability. We develop a test for orbital stability of relative equilibria using group invariant Liapunov-like functions. A variant of Ortega's result ([Ort98]) on formal stability of relative equilibria of conservative systems with symmetry can be obtained by applying this stability test to the energy-momentum functional. We then give a new proof of components of the Lagrangian block-diagonalization procedure ([Lew92]), which is an effective tool for the determination of formal stability. The new proof differs from the original in that we directly analyze the subspaces of the kernel of the differential of the momentum map, rather than constructing the so-called locked vector field, which determines a section of the tangent bundle that “almost” lies in the level set of the momentum map.; The second part is concerned with bifurcation from symmetric relative equilibria to asymmetric relative equilibria, using the isotropy subalgebra as the bifurcation parameter. We derive a bifurcation test by using the Local Linearization Theorem, Liapunov-Schmidt Reduction, and the Equivariant Branching Lemma. This test was inspired by a similar test in [Lew93]. The new test lays out conditions under which the Liapunov-Schmidt reduction can be carried out, and the Equivariant Branching Lemma can successfully be used to locate symmetry-breaking bifurcation. As a result, no prior knowledge of these tools is needed for the application of the test. Discrete symmetries play crucial role in the implementation of the bifurcation test and stability analysis. In our examples, reflections in O(2)) are the pertinent discrete symmetries.; In the third part, we analyze a system that we call the pseudo-Lagrange top. It is like a Lagrange top, except that the top is linearly deformable. Abundant use is made of the representation of the isotropy subgroup in the process of applying the methods presented in the earlier chapters to test for the bifurcation and compute the formal stability of symmetric relative equilibria.