Investigation Of Some Problems About The Symmetry And Conserved Quantity Of Dynamical Systems

The present dissertation treats Lie symmetries and conserved quantities for dynamical systems, including the Lie symmetries and Hojman's conserved quantities of dynamical systems, the inverse problem of Lie symmetries and conserved quantities, and the variational principle of discrete dynamical systems and the discrete Noethers theorem. The dissertation consists of six chapters.The first chapter surveyed briefly the resent progresses in the theory of non-Noethers conserved quantities, the inverse problem of Lie symmetries and conserved quantities, and the symmetries and conserved quantities of discrete mechanical systems.Chapter two proposes the unified form of Hojman's conservation law and Lutzky's conservation law. Firstly, the author introduces the general Lie group of transformations that the variations of both the time and the generalized coordinates are considered, derives the determining equation of Lie symmetry for the system, presents a new conservation law, which contains the Hojman's and the Lutzky's conservation law as two special cases, and obtains a condition to exclude trivial Hojman's conserved quantities. Next, the author investigates the Lie symmetries and Hojman's conserved quantities for the Birkhoff systems and the nonholonomic systems respectively. Finally, the author discusses the relation between Mei's symmetry and Lie symmetry for Hamilton systems, and develop a method to find Hojman's conserved quantities by using Mei's symmetries.Chapter three analyzes the inverse problem of Lie symmetries and conserved quantities for dynamical systems. Firstly, the author generalizes the method used to find the characteristic functional structure of velocity-dependent infinitesimal symmetry transformations for second order differential equations by Katzin and Levine, and studies the characteristic functional structure of infinitesimal symmetry transformations for the first order nonholonomic constrained systems and the Birkhoff systems respectively. Next, the author studies the connection of first integrals with particular solutions of the nonsimultaneous variational equations for nonholonomic systems, and presents a new approach to find the inverse problem of Lie symmetry for nonholonomic systems.Chapter four deals with the symmetries and first integrals of discrete mechanics in...