The Theory Of Activity Frame And The Construction Of Differential Invariants Based On Activity Frame

The moving frames originated by physical and mechanical was first introduced by Caston Darboux, mainly to study the rigid motion of the subject. Then French mathe-matician E.cartan proposed moving frames theory, which was widely acknowledged for studying geometric properties of the submanifold under the action of a particular trans-formation group. Recetenly, along with the development of nonlinear science, Peter olver proposed equivalent moving frames theory to solve the limitations of the previous moving frames theory and provided a powerful computation tool for investigating the equivalence and symmetry properties of submanifolds under general Lie group actions, and made this theory more general and complete. That theory to be a powerful tool to solve the differ-ential equation, the basic symmetry problem, polynomial equivalent problem, classical invariant theory and its related application. Compared with E.carten’s moving frames theory, Peter Olver’s theory has a new improve.That is the recurrence relations, which complete prescribe the structure of the differential algebra and reduce the complexity of computation.Klein’s geometry view was put forward to all the different geometry properties, and regard them as the invariants under the action of group. Especially the invariant under the action of the Euclidean group, Affine group and Projective (transform) group can efficiently apply in many mathematical physics problems. So we can solve such problem in the aspect of differential invariants based on the moving frames. So the core content of this paper is to detail how to construct differential invariants by useing moving frames.In this paper, we start from the basic concepts of differential geometry, and accord-ing to Lie transformation group, lie algebra and the prolongation, and with the help of equivalent moving frames theory to study the basic structure method of the classical e-quivalent moving frames and its improved recursive method and differential invariants of the complete system, at last, their applications in mathematical physics problems is given.In this thesis, our work focus on the following five parts:Part 1 a brief introduction to the development course of moving frames and differen-tial invariants, and the exploration of equivalent moving frames theory developed by Peter Olver and its latest progress, also list the moving frames theory in mathematical physics problems in applications.Part 2 discusses the content of the two parts. Firstly, we introduce the relevant theories and concepts of manifold, vector field, differential form in differential geometry. Secondly, we introduce the basic concepts of lie group, lie transformation group, group action, variables function of built on the group and Maurer-Carten form, this part aims to provide a theoretical support for establishing a moving frames.Part 3 includes three aspects. First of all, we elaborate the basic definition and the structure algorithm of equivalent moving frames. Secondly, with the aid of the total derivative, prolongation and the infinitesimal generator, and normalize invariant process and prolong the infinitesimal generated, we get differential invariants under the action, invariant differential operator and syzygies. Finally, we use contact form to get differential invariants under the action, invariant differential operator and syzygies by the improved recurrence method of moving frames.Part 4 briefly introduces two applications of moving frames and differential invari-ants. The first application use the rotation transformation group and the equivalent affine group as an example to demonstrate the characteristic of the signature curve with the aid of symbolic software Maple, the second application is KdV equation, we demonstrate symmetry analysis of the differential equation.Part 5 summarizes the full text of the content, and provide an outlook of the future of the differential invariants and moving frames in application development.