The geometry of homogeneous spaces with application to Hamiltonian mechanics

Abstract

In this thesis we study some of the properties of homogeneous spaces. We are more interested in homogeneous spaces which are also manifolds. We have shown that homogeneous spaces are basically quotient spaces. Working with quotient spaces, we pushed to symplectic quotient, the Marsden-Weinstein-Meyer quotient or the symplectic reduction using an equivariant momentum mapping. We have shown that the reduction can be performed using an a ne action of a Lie group G on the dual g of the Lie algebra g using the momentum with cocycle . In this direction we also proved that a Riemannian structure on a symplectic manifold can be induced to the symplectic quotient through a Riemannian submersion. We have proved that if G is a compact, connected and semisimple Lie group, acting transitively on its Lie algebra g by the adjoint representation, and acting transitively on the dual g of its Lie algebra by the coadjoint representation, then there is a symplectic di eomorphism between the homogeneous space g=G, the adjoint orbit of the adjoint action and the homogeneous space g =G, the coadjoint orbit of the coadjoint action. We have extended this result to the applications to Hamiltonian mechanics and have shown that Hamiltonian vector elds on symplectic manifold lift to Hamiltonian vector elds on the cotangent bundle of this manifold. On the way to this result we have written equations of Hamiltonian systems using the deformed Poisson bracket and have proved that many properties of Hamiltonian systems with canonical Poisson bracket also hold with a more general structure, the deformed Poisson bracket.