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Dissertation zugänglich unter
Equivariant Transversality Theory Applied to Hamiltonian Relative Equilibria
Äquivariante Transversalitätstheorie angewandt auf relative Ruhelagen in Hamilton-Systemen mit Symmetrie
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Freie Schlagwörter (Deutsch):
Hamilton-Systeme , Symmetrie , relative Ruhelagen , Bifurkationstheorie , äquivariante Transversalitätstheorie
Freie Schlagwörter (Englisch):
Hamiltonian Systems , symmetry , relative equilibria , bifurcation theory , equivariant transversality theory
Lauterbach, Reiner (Prof. Dr.)
Tag der mündlichen Prüfung:
Kurzfassung auf Englisch:
In this thesis, we investigate the structure of relative equilibria in Hamiltonian systems with symmetry. We consider proper Hamiltonian group actions with a focus on compact groups. The new results are obtained from applications of equivariant transversality theory. The method is similar to the one developed by Michael Field and others applying equivariant transversality theory to bifurcation theory.
This is done in two different ways. The first application is a more general formulation of a famous result due to George Patrick and Mark Roberts valid for free group actions. In their article “The transversal relative equilibria of a Hamiltonian system with symmetry” (Nonlinearity 13) from 2000, they give a generic transversality condition for relative equilibria and deduce that the relative equilibria generically form a Whitney stratified set whose strata consist of relative equilibria of the same conjugacy class of the intersection of the isotropy groups of the momentum and the generator. We observe that Patrick’s and Roberts’ condition is a special case of a condition formulated in terms of equivariant transversality theory. In this way, their result can be partly generalized to non-free actions.
The second application is an investigation of the local structure of relative equilibria near an equilibrium at the origin in symplectic representations with compact symmetry groups. We first investigate torus representations and obtain that generically the topological structure of the relative equilibria coincides with that of the linearized vector field. In the generic situation, the set of relative equilibria near the origin is a union of manifolds that are tangent to sums of weight spaces with linearly independent weights of the centre space of then linearization. Conversely, there is such a manifold for any set of linearly independent weights.
These results can be applied to representations of a general connected compact group by restricting the action to the maximal torus. In this way, we do not obtain the whole generic structure of the relative equilibria, but we still predict branches that generically exist: Generically, the real parts of the sums of the eigenspaces of the linearization for each pair of purely imaginary eigenvalues are irreducible symplectic representations. They may be regarded as irreducible complex representations. Consider the set of weights of one of these irreducible representations. For each affine subset of the dual of the Lie algebra of the maximal torus that contains only a linearly independent subset of these weights (counted with multiplicity), there is a manifold of relative equilibria tangent to the sum of the corresponding weight spaces. Moreover, if we join subsets of weights of these kind of different eigenspaces and the union is linearly independent, we generically obtain a manifold tangent to the sum of the corresponding weight spaces, too.