Researches On Normal Form Of The Generalized Hamiltonian System

The normal form of dynamical system is one simplified system of the original sys-tem achieved by selecting good near-identity transformations. In this paper, the normal form of the generalized Hamilton system on m dimensonal Poisson manifold Rm and its computation are investigated. The generalized Hamilton system is the direct, general-ization for the classical Hamilton system defined on R2n. The dimension of phase space of classical Hamilton systems must be even number, but that of generalized Hamilton system may be any postive integer and even infinite. They describe a. variety of nonlinea.r dynamical models, including celestial mechanics, biological sciences, ion physios and mag-netohydrodynamics. The phase space of the generalized Hamilton system is a Poisson manifold with foliation structure and each leaf is an invariant symplectic. submanifold of one generalized Hamilton system confined on which the generalized Hamilton system is the classical Hamilton system on symplectic manifold.In this research, the generalized Hamilton system has the following form x=J(x)▽H(x), or,dxi/dl=∑jm1Jij(x)(?)H/(?)xj(x),i=1,...,m., where the skew matrix J(x)=(Jij(x))m×m is known ai Poisson structure matrix stisfied Jachbiidentity.H(x) is know as Hamiltoni-an. When every structure element Jij(x) is the linear homogeneous function with respect to x, the Poisson structure is called Lie-Poisson structure, which is isomorphic with Lie algebra structure.The theory and application of generalized Hamilton systems with Lie-Poisson struc-ture make a decisive role in practical researches. According to existed researches, when J(x0)=0and the linearized part J1(x-x0) of J(x) at x=x0is simisimple, there exists reversible transformation mapping the Poisson structure J(x) to J1(x-x0) in the neighbor of x=x0.In Chapter1and2, the research background and preliminaries are introduced. In Chapter3, the normal form of classical Hamiltonian systems and their computation are extended to the generalized Hamilton system with Lie-Poisson structure. By utilizing the flow defined by solution of generalized Hamilton system as the near-identity transforma-tion, finding normal form of the generalized Hamilton system is converted to simplifying Hamiltonian, because such transformation preserves the Lie-Poisson structure of the gen-eralized Hamilton system. By expanding H(x) as series H(x)=H1(x)+H2(x)+ where Hk(x) is a k+1order polynomial, and defining the adjoint operator adH1={·,,H1) and detailedly analyzing homological equation Hk=Hk+adH1(Wk), the condition of the normal furm H of Hamiltonia is achieved and the algorithm and formula of generating function W of near-identity transformation are determined.In order to illustrate theory and methods proposed in Chapter3, one three-dimensional generalized Hamiltonian system x=J(x)▽H(x) with unitary Lie group structure, the corresponding Lie-Poisson structure matrix is Hamiltonian is the form of series of H(x)=H1(x)+H2(x)+…, whereH1(x)=a1x1+a2x2+a3x3The normal form of x=J(x)▽H(x) is obtained as x=J(x)▽H(x).(1) When a1a3=0, wo have:(2)Whena1a3≠0, we have:Finally, by employing knowledge of Hamiltonian system and detailedly analyzing the phase trajectory structure and stability of equilibria of truncated system of normal form system, orbit diagrams of different two-dimensional symplectic leaves in phase space are achieved.