Basic Theory For Dynamics Of Fractional Generalized Hamiltonian Systems

The research history of fractional order differential equation has more than300years, scientists mainly established three kinds of fractional operators: Riemann-Liouville fractional order operator, Caputo fractional order operator and Riesz fractional order operator. In1980’s, scientists found there were a large number of fractal dimension examples in nature. This discovery greatly aroused the interest of scientists who wanted to explore fractional order equation. Fractional order differential equation system in dynamic, the non-Newton fluid mechanics, biological physical and biological mechanics, etc have a good application.In recent20years, scientists have established the fractional order Lagrange mechanics, fractional order Hamilton mechanics, fractional order generalized mechanics and fractional order nonholonomic dynamics theory in fractional dynamics. Generalized Hamilton mechanics is also an important kind of classical mechanical system, in a lot of science and engineering, plays a foundation role. But, fractional order generalized Hamilton mechanics has not been set up, including the fractional order generalized Hamilton equations, gradient representation, integral theory, the symmetry theory, symmetry perturbation theory, the balance stability and the movement stability theory, etc.This paper presents the fractional order generalized Hamilton dynamic equations, research gradient representation, algebraic structure, Poisson integral, variational equations and integral invariants of the fractional order generalized Hamilton system.In the first chapter, we study briefly the latest progress in the theory of the fractional order dynamics and the generalized Hamilton dynamics.In the second chapter, the author researches three different definitions of fractional order derivatives. By using the idea of the generalized Hamilton equations, the author establishes the sequence fractional order generalized Hamilton equations based on Riemann-Liouville, Caputo and Riesz fractional order derivatives, respectively. In chapter three, firstly, we establish the a fractional order generalized Hamilton equations based on Riemann-Liouville, Caputo and Riesz fractional order derivatives respectively. Secondly, as the special case of the a fractional order generalized Hamilton equations, we obtain the conditions on which a a fractional order generalized Hamilton equations can be degenerated as a generalized Hamilton equation, a fractional Hamilton equation and a Hamilton equation.In chapter four, we study the conditions on which a α fractional order generalized Hamilton equation can be a gradient system or a second-order gradient system, and give the gradient representation of the system. As the special case, we get the conditions on which generalized Hamilton equations, fractional Hamilton equations and Hamilton equations are a gradient system or a second order gradient system, and obtain the corresponding gradient representation.In chapter five, the author studies algebra structure and Poisson integral, presents seven integral theorems of fractional generalized Hamilton equations. As the special case, the author obtains algebra structure and Poisson integral of generalized Hamilton equations, fractional Hamilton equations and Hamilton equations.In chapter six, we study variational equation of fractional generalized Hamilton equations. By using variational equation and first integral, we construct a kind of integral invariants. Then, as the special case, we obtain variational equation and integral invariants of generalized Hamilton equations, fractional Hamilton equations and Hamilton equations.In chapter seven, the author concludes the dissertation by summarizing the results and presents several ideas in the further work.