Stability Of The Equilibrium Of Finite-dimensional Autonomous Hamiltonian System

If the quadratic part of Hamiltonian at equilibrium points is sign-definite,all the system has extremum at equilibrium points,then all the system is sta-ble in the sense of Lyapunov.And if the Hamiltonian system isn't spectral stability,it is unstable.Therefore spectral stable Hamiltonian systems at e-quilibrium points,becomes Lie normal form by Lie transformation algorithm,then we must analyse the higher order term of Hamiltonian to analyze Hamil-tonian system in the stability of the equilibrium point.The paper mainly uses the examples of the analyse of stability of equilibrium points for the plane circular restricted three-body problem to analyse the stability of equilibrium points of autonomous hamiltonian systems.We can find the stability of the system is related to the correlation of its linear partial eigenvalues in the ratio-nal domain.If it is linear dependent,we can say it has resonance relationship,while there isn't the resonance relationship,the first integral of the system at the equilibrium point is positive definite,the system is stable at the equilib-rium point.The lie stability of resonance exists only in special cases.The paper gives some criterions of stability of autonomous hamiltonian systems at equilibrium points when there exists single resonance.The paper is structured in five chapters:In Chapter 1,introduce the background and development of the stability of finite dimension,autonomous Hamiltonian systems in the sense of Lyapunov;The Chapter 2 gives the crite-rion of linear stability and spectral stability,to finite dimensional autonomousHamiltonian systems;And Chapter 3 gives the steps of transforming original Hamiltonian systems at equilibrium points to the simplest by lie transforma-tion;Then in the Chapter 4,judge the stability in the sense of Lyapunov and lie stability,when linear parts are diagonalizable or the system has two-order single resonance;In the last Chapter,the first part by the criterions of the stability of equilibrium points for the plane circular restricted three-body problem,summarize the classical theorems of the stability,especially for the resonance exist,in the second part,in different cases and the methods to deter-mine the linear stability of a planar three-body problem of lagrangian solution by perturbation theory for linear operators,?-index theory of symplectic paths and the variational iteration.