Singularity theory and geometry in the motion of a top

The aim of this thesis is to examine the spinning top from the point of view of the Smale programme for studying mechanical systems with symmetry. This programme consists of finding the global topological structure of the map E x J : TM -. R x P * where E is the total energy of the system, J its momentum mapping, which in our case is just its angular momentum. TM is the phase space and Q * is the P dual of the Lie algebra of the Lie group G which acts on the configuration space M producing the symmetry. We are here concerned with examining the nature and configuration of the singularities of this and related maps using the machinery of ){ and .q equivalence and of finite determinacy. We are able to interpret various types of motion of the top in terms of singularities and their unfoldings. Of particular importance isthe subset of TM corresponding. to steady precession whose corresponding geometry in the cotangent bundle we exhibit explicitly.