The Research Of Symmetry And Conserved Quantity For The Constraint Mechanical Systems With Time Delay

In this dissertation we mainly study Noether's symmetries and the corresponding conserved quantities of the continuous non-conservative systems with time delay and the holonomic conservative systems on time scales with time delay.At the beginning, we present some basic definitions and properties about time scales in order to lay a foundation for the analysis below. Firstly, we introduce a series of definitions, such as time scales, the jump operator, chain functions and nabla derivative on time scales etc. Then, we discuss the related properties and a series of conclusion on time scales are obtained. The definition of integral and its properties is given. At the end of this chapter, we present some properties of calculus of variations on time scales which include the exchanging relationships between the isochronous variation and the nabla derivatives, the exchanging relationships between the isochronous variation and backward jump operator and the relationships between the isochronous variation and the total variation on time scales. For each equation, the related proof is given to illustrate its validity.Then, we discuss the Noether's symmetries and conserved quantities of the general continuous non-conservative constrained mechanical systems with time delay. Noether's symmetry theories of non-conservative mechanical systems covering in both variational and optimal control problems with time delay is obtained. Firstly we prove Euler-Lagrange equation and DuBois-Reymond necessary condition with time delay for non-conservative mechanical systems, then we give non-conservative Noether's theories with time delay based on the generalized quasi-invariance of Hamiltonian action of the systems under the infinitesimal transformations with respect to the time and generalized coordinate. Secondly we introduce the holonomic non-conservative system in optimal control and establish the non-conservative Hamiltionian system. the Maxima criterion and the more general DuBois-Reymond necessary optimality condtions are proposed. Lastly, Noether's symmetry theories of the optimal control problems for non-conservative systems with time delay based on the quasi-invariance of Hamiltonian action of the systems under the infinitesimal transformations with respect to the time and generalized coordinates are obtained.Finally, we study the Noether's theorem of the Lagrange conservative mechanical systems with time delay on time scales. Given the special time scales, the variation fundamental formula of the Hamilton's action is presented for non-conservative systems with time delay on time scales by means of calculation. We define Noether's symmetry and the quasi symmetry of the Lagrange mechanical systems with time delay on time scales. And then we obtain the criterion of symmetry and quasi symmetry and acquire the Noether's equation. Eventually Noether's conserved quantity of the Lagrange conservative mechanical systems with time delay on time scales is obtained.Innovative points in this thesis:Noether's symmetry theories of the continuous holonomic non-conservative systems with time delay and the optimal control problems are obtained; Noether's theorem for holonomic conservative systems with time delay on the given time scales is obtained.