摘要: Major work on the dynamics of coupled rigid bodies was done. We studied both the case of three coupled rigid bodies in the plane, with a complete stability, bifurcation, and chaotic solutions analysis, but also studied the three d... 展开 Major work on the dynamics of coupled rigid bodies was done. We studied both the case of three coupled rigid bodies in the plane, with a complete stability, bifurcation, and chaotic solutions analysis, but also studied the three dimensional case. In the energy-momentum method, for mechanical systems with Hamiltonian H of the form kinetic energy (K) plus potential (V), a way was found to choose variables that makes the determination of stability conditions sharper and more computable. The poisson brackets of free boundary fluid equations has been determined. In the homogeneous case, it has been shown already that the structure of the bracket is that of a Yang Mills theory for the principal bundle whose total space consists of the embeddings of a given domain in space, the base space is the space of unparametrized fluid shapes and the group os the particle relabeling group. The geometric reasons for the integrability of the planar three point vortex motion is terms of dual pairs appearing in the study of the geometry of Poisson manifolds has been given. The study of the hydrodynamic bifurcations was begun on the example of a rigidly rotating incompressible homogeneous disk. (JHD) 收起