Normal Form Theory And Application Of Linear Hamiltonian System

The normal form of symplectic matrix provides the basis for studying the topological structure of symplectic group and its subsets,and the index theory of Maslov type and Hamiltonian system.In the research of index theory and its iteration theory of Hamiltonian system,the similar normal form of symplectic matrix plays a key role.More and more experts and scholars have carried out in-depth research on such problems.In this paper,we start to summarize the method of teacher long Yiming to deduce the normal form of the symplectic matrix of the eigenvalue on the unit circle and outside the unit circle.Then,the symplectic matrix is corresponding to the normal form of the equilibrium point of the Hamiltonian function,and the basic normal form of the equilibrium point of the Hamiltonian system is given by the basic normal form of the symplectic matrix,and its Krein form is further summarized.The anti symplectic background appears naturally when there is time to invert symmetry.Based on the study of symplectic matrix,the properties of anti symplectic matrix are given and the normal form of anti symplectic matrix is derived in detail.As an application,Maslov index decomposition formula with time inversion symmetry and the stability of periodic orbits are studied.This paper is divided into five chapters:the first chapter introduces the background and development of symplectic matrix normal form and index theory;the second chapter deduces symplectic matrix normal form,and gives the normal form of symplectic matrix whose eigenvalue is-1,eigenvalue is 1,eigenvalue is on unit circle and eigenvalue is on unit circle;the third chap-ter summarizes the normal form of Hamilton system equilibrium point and gives the Hamilton system equilibrium point.In Chapter 4,the properties of the anti-symplectic matrix and the derivation of the normal form of the anti-symplectic matrix are given.In Chapter 5,the decomposition formula of the Maslov index and the conclusion of the stability of the periodicorbit are given.