Study About The General Construction For Symmetry And Operators Of Infinite Dimensional Hamiltonian System

In the contemporary society,vigorous developments have been made in the scien-tific research,especially in the Hamiltonian system,which has also played an important role.Many models in research fields,such as Classical mechanics,Celestial mechanic-s,Aerospace Sciences and Bioengineering,appear in the form of Hamiltonian system because of its special and concise structure,thus promoting the progress and devel-opment of science.The article puts forward the new research ideas and obtains the formal solutions of symmetry and the general structures of the matrix Hamiltonian operators.In the first chapter,the paper firstly introduces the research object;secondly,the generation and development of symmetry and infinite-dimensional Hamiltonian oper-ator are reviewed after consulting plenty of literature;then the difficulties of previous studies confronted are expounded;finally,the research ideas and the main achieve-ments are given.In the second chapter,we propose the new ideas for obtaining the solutions of symmetry under infinite dimensional linear canonical Hamiltonian system:using vec-tor representation method and the sum of differential operators,linear Hamiltonian canonical system is well converted into a certain type of differential equations.Then the symmetry determining equations are presented according to infinitesimal criteri-on.And thus the new conclusions are obtained in Hamiltonian system with constant and variable coefficients respectively.Meanwhile,some related examples are given to illustrate the application of the results.In the third chapter,further exploration of the construction about infinite di-mensional matrix Hamiltonian operators and Hamiltonian pairs is made.Firstly,by constructing three forms of operator,the conditions making them into Hamiltonian operators are deduced;then new Hamiltonian operators are constructed using these conditions;meanwhile,the correctness and generality of the conclusions are verified.Furthermore,several Hamiltonian pairs are constructed.Finally,the paper summarizes the research work briefly,points out the deficiency and describes the future work we are going to study further.