Generalized Hamilton System Dynamics Study Of The Basic Theory

The generalized Hamiltonian dynamics systems have been played an important role not only in science, but also in technology, and received widespread attention from academics. The paper we have studied integral theory, symmetry and conserved quantity, symmetrical perturbation and adiabatic invariant, stability for motion of generalized Hamiltonian system.In the first chapter, we review the history and the current situation of studies on generalized Hamiltonian system, some problem of dynamics of constrainted systems, and some problems of generalized Hamiltonian system for their existence.In the second chapter, we study integral theory of generalized Hamiltonian system. The method of construction of an integral invariant is studied, then the relation between the integral invariant and first integral is discussed, finally we studies the integral invariant and first integral of three groups of Volterra model, three-body problem in the three vortex plane relative motion and Euler equations of rigid body round fixed-point motion.In the third chapter, we study the conserved quantity directed by symmetry of generalized Hamiltonian system. Based on the invariance of equations of motion for the system under general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations and conserved quantities of the generalized Hojman are given, the form of conserved quantities and the conditions for their existence are obtainted. Then, we proofed the Mei symmetries of generalized Hamiltonian system can directly lead to a new type of conserved quantities, and given the form of conserved quantities and the conditions for their existence. Finally, using the method of Mei symmetries of generalized Hamiltonian system, we obtain a new type of conserved quantities of three-body problem in the three vortex plane relative motion.In the fourth chapter, we study adiabatic invariants directed by symmetrical perturbation of generalized Hamiltonian system. We found disturbed equations of generalized Hamiltonian system. Based on the invariance of equations of motion for the system under general infinitesimal transformation of Lie groups, the Lie symmetrical perturbation determining equations and adiabatic invariants of the generalized Hojman are given, the form of adiabatic invariants and the conditions for their existence are obtainted. Finally, one example is given to illustrate the application of the method and result.In the fifth chapter, we study the stability for manifolds of equilibrium states. Based on Lyapnuov's stability theories, equilibrium equations, perturbation equations and first approximate equations of generalized Hamiltonian system are given. Then we study the stability for manifolds of equilibrium states of the autonomous generalized Hamiltonian system and give three propositions. Further, we studies the stability for manifolds of equilibrium states of three-body problem in the three vortex plane relative motion, Euler equations of rigid body round fixed-point motion and three groups of Volterra model.In the sixth chapter, we present some ideas for the future researches.