Title:

Local dynamics of symmetric Hamiltonian systems with application to the affine rigid body

This work is divided into two parts. The first one is directed towards the geometric theory of symmetric Hamiltonian systems and the second studies the socalled affine rigid body under the setting of the first part. The geometric theory of symmetric Hamiltonian systems is based on Poisson and symplectic geometries. The symmetry leads to the conservation of certain quantities and to the reduction of these systems. We take special attention to the reduction at singular points of the momentum map. We survey the singular reduction procedures and we give a method of reducing a symmetric Hamiltonian system in a neighbourhood of a group orbit which is valid even when the momentum map is singular. This reduction process, which we called slice reduction, enables us to partially reduce the (local) dynamics to the dynamics of a system defined on a symplectic manifold which is the product of a symplectic vector space (symplectic slice) with a coadjoint orbit for the original symmetry group. The reduction represents the local dynamics as a coupling between vibrational motion on the vector space and generalized rigid body dynamics on the coadjoint group orbits. Some applications of the slice reduction are described, namely the application to the bifurcation of relative equilibria. We lay the foundations for the study of the affine rigid body under geometric methods. The symmetries of this problem and their relationship with the physical quantities are studied. The symmetry for this problem is the semidirect product of the cyclic group of order two Z2 by 50(3) x 50(3). A result of Dedekind on the existence of adjoint ellipsoids of a given ellipsoid of equilibrium follows as consequence of the Z2 symmetry. The momentum map for the Z2 x, (50(3) x 50(3)) action on the phase space corresponds to the conservation of the angular momentum and circulation. Using purely geometric arguments Riemann’s theorem on the admissible equilibria ellipsoids for the affine rigid body is established. The symmetries of different relative equilibria are found, based on the study of the lattice of isotropy subgroups of Z2 x, (50(3) x 50(3)) on the phase space. Slice reduction is applied in a neighbourhood of a spherical ellipsoid of equilibrium leading to different reduced dynamics. Based also on the slice reduction we establish the bifurcation of Stype ellipsoids from a nondegenerate ellipsoidal equilibrium.
