Theoretical Reserch Of The Symmetry Of Dicrete Mechanical System In Phase Space

In morden theory of symmetry, the corresponding conserved quantities of the system are obtained mainly through the symmetry analysis of the system. The existence of conserved quantities is very important to understand the physical state and properties of the system. The researches on the thoery of symmetries and conserved quatities of discrete mechanical system have important theoretical value and practical significance. In this paper, we study the symmetris and their resulting conserved quantities of discrete mechanical systems in phase space with the difference discrete variational approach of variable time step, based on the thoery of symmetries and conserved quatities of continuous mechanical system. Firstly, we derive the dynamics equations, which include discrete canonical equations and discrete energy evolution eqution, of the discrete holonomic system, the discrete nonholonomic system and the discrete mechanical system with variable mass in phase space, from difference discrete action principle. Secondly, we study the Noether symmtry of the discrete holonomic system, the discrete nonholonomic system and the discrete mechanical system with variable mass in phase space. The discrete Noether equations and their resulting discrete Noether conserved quantities of the three kinds of systems are given. Then, the Mei symmetry of the discrete holonomic system, the discrete nonholonomic system and the discrete mechanical system with variable mass in phase space are studied. The discrete determining equation of Mei symmetry ( discrete Mei equation ) and discrete Mei conserved quantities led by Mei symmtry of the system are given. Finally , we study the Noether-Mei unified symmetry of the discrete holonomic system, the discrete nonholonomic system and the discrete mechanical system with variable mass in phase space. The discrete determining equations of the Noether-Mei unified symmetry of the systems are got. The conditions for the Noether conserved quantities and the Mei conserved quantities leb by the unified symmetry and the forms of the conserved quantities are discussed.