Studies On The Generalized Hamiltonian System With The Structure Of Lie Algebra A_4,1

The Hamiltonian systems with Lie-Poisson structure corresponding to Lie algebra structure exist widely in a variety of practical fields, particularly celestial mechanics, plas-ma physics, space science and bioengineering. Hamiltonian systems and their perturbed systems are found in these fields. Recent researches indicate that these systems have a close connection with optimal control problems on Lie group. For Hamiltonian system with Lie algebra structure, three dimensional Lie algebra has the well-known Bianchi clas-sification. Also, the persevering structure transformation of three dimensional Lie algebra, simplified generalized Hamiltonian systems and their dynamical behaviors are studied in many publications. However, for the generalized Hamiltonian system with higher dimen-sional Lie algebra structure, no general results are reported. In this manuscript, the following quadratic homogeneous generalized Hamiltonian system with four dimensional Lie algebra is analyzed, where the structure matrix J(x) is the Lie-Poisson structure matrix corresponding to four dimensional algebra A4,1[13] Hamiltonian function H(x)=1/2xTSx is a quadratic homogeneous function and S is a sym-metric matrix with ten parameters. In order to investigate dynamical behaviors of the generalized Hamiltonian system with ten parameters, the quadratic homogeneous Hamil-tonian is classified into13equivalence classes by employing reversible linear structure-preserving transformations. Only the first class contains two independent parameters, and others are involved with at most one independent parameter and their orbit struc-tures in phase space are simple. The generalized Hamiltonian system corresponding to the first class Hamiltonian function is The rest of this manuscript detailedly analyze this two-parameter system. Equilibriums of this system are achieved and their bifurcation and stability are studied. All orbit structures on different foliates are found. The explicit solutions of homoclinic orbits, heterclinic orbits and periodical solutions of the system are determined. Finally, by using the perturbation theory of generalized Hamiltonian systems, the periodical perturbed system of this four dimensional generalized Hamiltonian system is studied. Next, by computing Melnikov functions of corresponding orbits, conditions of existence of periodic orbits and homoclinic orbits of perturbed system are provided.In Chapter1, research background and preliminary are introduced. The structure preserving transformation matrix of the four dimensional Lie algebra A4,1is studied and the quadratic homogeneous Hamiltonian function with10parameters is simplified into13classes of Hamiltonian function containing at most two independent parameters by employing structure preserving transformation and properties of generalized Hamiltonian system. Thereby generalized Hamiltonian system corresponding to A4,1is simplified into13classes. In Chapter3, dynamical behaviors of the first class in achieved13classes, including equilibriums, their stability, bifurcation, and orbits on phase space are analyzed and explicit solutions of homoclinic obits and heterclinic orbits are achieved. Moreover, dynamical behaviors of other12classes are analyzed. In Chapter4, conditions of existence of homoclinic orbits and periodical orbits of perturbed system of the first class are discussed.