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The Study and Application of the Generalized Hamiltonian System with Spherical Foliation Structure

Author: LiZhuo
Tutor: ZhaoXiaoHua
School: Zhejiang Normal University
Course: Applied Mathematics
Keywords: Gyrostat Foliation Bifurcation Exact Solution Melnikov Method
CLC: O175
Type: Master's thesis
Year: 2011
Downloads: 3
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The thesis deals with the dynamic behavior and application for a class of 3-dimensional generalized Hamiltonian systems by using the method of dynamical sys-tem. The phase space of such system is of a spheral foliation structure, of which the sphere with different radiuses is foliation (invariant manifolds), on which the portraits of the system lie. This class of system has been widely applied in many physical areas, such as mechanical engineering, celestial mechanics, optics, and molecular dynamics. It is of theoretical and pragmatic importance to study and understand more about the properties of various solutions to the attitude of a triaxial gyrostat in torque-free motion with three rotors and the rotational dynamics of H2X molecules.Due to the complication of the nonlinear issues, scientists pay more attentions to the equilibrium solution, periodic solution, homoclinic and heteroclinic solution, focusing on the existence and stability of such solutions. Some special cases of the complete bifurcation problem and the stability of the system have been studied with the given parameters.The thesis mainly concerns the model of a gyrostat in torque-free motion which is a class of 3-dimensional generalized Hamiltonian systems with spheral foliation struc- ture that can be written as ordinary differential equations: The corresponding Hamiltonian can be written as the following equation: H=1/2(α1g12+α2g22+α3g13)-(α1g1f1+α1g2f2+α3g3f3) (4)Under the background of the motion for gyrostat, the parameter gi is the angular momentum vector, ai is the inverse of the principle moment of the inertia,fi is the angular moment of the rotors. Moreover, the norm of the angular momentum vector is an integral.‖G‖= g12+g22+g32= constant, (5) Thus, the generalized Hamilton system have two first integrals:Hamiltonian H and the norm of the angular momentum vector(Casimir function). Therefore, the system could be reduced as 1-dimensional integrable system on the spherical foliation structure S2(G). S2(G)={(g1, g2,g3)(?)g12+g22+g32=L2= constant}The integral of the system assures that the portraits lie on the sphere with fixed radius, which is determined by the initial values of the variables.Based on different parametric conditions, we classify the triaxial gyrostat in torque-free motion into five different types. Previous researches have devoted to some of the types and analyzed the dynamic features, such as the number and stability of the equi-libriums. According to the theory of generalized Hamiltonian system, the thesis discusses the dynamics of two classes of gyrostat:(1) axially symmetric gyrostat (α1=α2>α3) with three rotors spinning about each principal axis of inertia (f1,f2, f3≠0); (2) asym-metric gyrostat (α1>α2>α3) with two rotors spinning about any two of the principal axes of inertia (fi, fj≠0, fk =0). In Chapter 2, we firstly introduce the basic defini-tions, important properties and related theoretical method of generalized Hamiltonian system. Meanwhile, we get the Hamilton structure of gyrostat model. In Chapter 3 and 4, we analyze the number and stability of the equilibriums in the complete para-metric regions. By using mathematical software Maple, we find out the global phase portraits on the unit sphere by the implicit function plotting technique. Furthermore, the orbits under various parametric regions can be expressed as exact solutions by Jacobi elliptic function or hyperbolic function. Based on this, Chapter 5 makes an analysis on planer homoclinic orbits and planer heteroclinic orbits of perturbed system by adopting perturbation theory. By calculating the responding Melnikov function, we notice that chaos may take place under small perturbation.

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