The Symmetries And Conserved Quantities Of Several Physical Problems

Nonlinear phenomena are an important branch in modern mathematics which has many applications in several aspects. Both in theory and practical applications, nonlinear partial differential equations are used to describe the mechanics, controlling process, ecological and economic systems, chemical cycle system and epidemiology and so on. The using of the nonlinear partial differential equations fully considers the impact of space, time and time lag, which is more accurate in solving the problems.Nonlinear partial differential equations are very widely used in mathematics, physics, engineering, biotechnology and other aspects. In general, such nonlinear equations are often very difficult to solve explicitly. Because of the complexity of the nonlinear mathematical physics itself, there are some special circumstances which make the problem difficult or not to solve with general method. The basic idea of solving these equations is transformation and reduction, which is simplify the complex equation. However, symmetry group techniques provide one method for obtaining such solutions of partial differential equations (PDEs).In this thesis, we mainly investigate Lie symmetries and conserved quantities of the two-dimensional nonlinear partial differential equation, and a new type of conserved quantity of Mei symmetry for the motion of the nonholonomic mechanical system and the mechanico-electrical coupling dynamical systems.The second chapter of the full text major discusses Lie symmetries and conserved quantities of the two-dimensional nonlinear diffusion equations. Based on the invariance of the two-dimensional nonlinear diffusion equations under the infinitesimal transformation with respect to the generalized coordinates and time, the determining equations of Lie symmetries are presented. The Lie groups of transformation and infinitesimal generators of these equations are obtained. The conserved quantities associated with the nonlinear diffusion equations are derived by integrating the characteristic equations. The third part investigates the Lie symmetry of a two-dimensional nonlinear Schr?dinger's equation. In the fourth and fifth parts, respectively, we get a new type of conserved quantity of Mei symmetry for the motion of nonholonomic mechanical system and the mechanico-electrical coupling dynamical systems under the infinitesimal transformations. The criterion of Mei symmetry for the two systems is given, and simultaneously, the condition of existence of the new conserved quantity of Mei symmetry for the two systems is obtained. Finally, examples are given to illustrate the application of the results.There are three novelties of this thesis: first, we investigate the Lie symmetries and conserved quantities of two-dimensional of nonlinear partial differential equations; Second, we get a new conserved quantity of Mei symmetry of the motion of nonholonomic mechanical system and the last is a new conserved quantity of Mei symmetry of the mechanico-electrical coupling dynamical systems.