Studies On Poisson Structure Matrices And Their Construction Methods

The studies in this thesis are related with the generalized Hamiltonian system(GHS) on an n-dimensional Poisson manifold, which has following form: or compact vector form x=J(x)·▽H(x). Where H(x) is a C1and real-valued function. Moreover, H(x) is a constant of the motion of the system, which plays the role of Hamiltonian. The matrix J(x)=(Jij(x)) is called Poisson structure matrix, the elements of which are C1and real-valued functions and satisfy the following two propertie(ⅰ) skew-symmetric:(ⅱ) Jacobian identities:The GHS, including classical Hamiltonian system as its special case, is an impor-tant field in the nonlinear sciences and widely applied in such a variety of disciplines as mathematics, life science and even social sciences. Especially. Hamiltonian system and its perturbed system are constantly employed to express models obtained from celestial me-chanics, plasma physics and biotechnology. The frame of GHS theory and many methods dealing with GHS have been developed.Hence, it is very significant to distinguish whether or how a given dynamical system defined by ODEs can be recast into the generalized Hamiltonian system, which amount to finding suitable structures matrix J(x) satisfying (2) and (3) and the Hamiltonian. This type of problem is still a challenging problem. Thus, the deeper study on the matrix solution J(x) satisfying (2) and (3) is a very important research topic. This thesis deals with this topics and is organized as follow. In Chapter1. we review previous works and background information on our topics. Some basic defini-tions.important properties and related theoretical methods of generalized Hamiltonian system are introduced in Chapter2. In Chapter3, some new methods of constructing three-order Poisson structure matrices are developed. In Chapter4, some new results on n-order (n≥4) Poisson structure matrices and their constructions are given and anal-ysed. In Chapter5, we discuss whether the congruent matrix of general structure matrix is also structure matrix, and obtain two new results.