Study On Some Problems Of Generalized Hamiltonian System And Poisson Manifold

This paper discusses some problems on Poisson manifold--thephase space of the generalized Hamiltonian system:1. Casimir functions on Poisson manifold2. An important case of weinstein's splitting therom3. A criterion of Poisson isomorphism4. Poisson structural group and its application5. Integration and integral invariants on exact Poisson manifoldThe five problems portray construction attributes of Poisson manifold from different aspects and angles Not only some new concepts and viewpoints are put forward but the values of their application in the theory of Hamiltonian system are clarifiedThe paper is divided into five parts (every problem is a part)In the first part, the algebric properties of the Poisson structure (Poisson tenser field)are investigated by using the characteristic function - casimir function Besides,two new formulars about Hamiltonian field are obtainedIn the second part, a special and important case of the splitting therom is posed and provedIn the third part, a necessary and sufficient condition of Poisson isomorphism is madeThe fourth part and the fifth part are the most important parts of the full text. In these two parts,something essential can be seen more clearly by using the new concepts introduced in In the fourth part,a new concept - "Poisson structural group" is put forward. To see the essence of geometry by studying it's tranformation group,it is,in fact,representing the spirits of"Erlanger programe". In the fifth part, A new model - "exact poisson manifold "is established. By using some other new concepts such as "characteristic manifold", "vortex line", the integration and integral invariants on "exact poisson manifold" is studied. Not only some new resoults are derived but some old resoults are generalized. For instance, Liouville's therom is generalized to"the exact poisson structure is invarant under the action of Hamiltonian flow-map".All these are expected to settle some basis for the further study of dynamatic system. This is the main purpose of this paper...