Obtaining Methods And Applications Of Recursion Operators And Conservation Laws In The Hainiltonian System

As is known to all, Hamilton system is a special partial differential equation system, which is also a important topic in the research of mathematics and physical science. Since1834, the system has been developed gradually into a research direction which is focused on by today’s scientists. In this paper, we discuss how to get differential equation’s new re-cursion operators and their conservation laws in the Hamiltonian system. Firstly, we apply the linearized method to obtain recursion operators in the Hamiltonian system, and validate the feasibility of the method, then we obtain conservation laws by the direct construction method and discuss its general form in the Hamiltonian system. We explain these conclu-sions through some examples and get some new results.Chapter Ⅰ, we make some investigation about the nature and the structure of the re-cursion operators, and present research status of the conservation laws, which helps us to do some more discussion about recursion operators and conservation laws.Chapter Ⅱ, we introduce about the obtaining methods and its applications of the re-cursion operators in the Hamiltonian system. It means that in the case of determining the general form of recursion operators, we get a determining equation group which is satisfied by these recursion operators, and win the new matrix form of recursion operators. In the process, we verify that in the Hamiltonian system, it also consist with the relation which is in the differential equation system; through the analysis, we gain more recursion operators in the Hamilton system than in the differential equation system; and we verify that it’s still recursion operators which is after linear combination in Hamilton system by some examples.Chapter Ⅲ, we obtain some new conservation laws by the direct construction method and get equations group about Conservation factors which is based on different forms of Hamiltonian operators. So we obtain new conservation laws of the Hamiltonian canonical equations. In this way, we can provide some ideas from researching the method of conser-vation laws.Chapter Ⅳ, we make a general summary in this paper, and make a primary plan in the terms of current shortcomings and inadequacies in our work.