The Investigation On Symmetry And Conserved Quantity Of Discrete Constrained Dynamical System

Using an infinitesimal Lie transformation group method, the symmetrical properties of the discrete constrained dynamical systems are investigated in this dissertation. Meanwhile, we employ symmetry analytical approach to exploring the discrete conserved quantities of the systems. In chapter one, the overview of the study on the symmetries and conserved quantities of constrained mechanical systems is presented, and the general definition of symmetry is given. Besides, a general discussion of the significance, the approach, the historical development, as well as the current research of the symmetries and conserved quantities for the continuous and discrete constrained systems, including the Noether symmetry, Mei symmetry, Lie symmetry, and several other symmetries, are developed. In chapter two, we investigate the dynamical equations of the discrete constrained systems. The total variational principle, including the time variational, is proposed. In addition, the dynamical equations and the constrained equations for discrete Lagrangian system, discrete Hamiltonian system, discrete non-conservative Lagrangian and Hamiltonian systems, discrete system with variable mass, discrete system with dependent variables, discrete nonholonomic systems with Chetaev also non-Chetaev type constrains, discrete system with unilateral constraints are constructed. The dynamical equations and the constrained equations are discrete Euler-Lagrange equations, discrete canonical equations of motion, discrete energy evolution equations, holonomic and nonholonomic discrete constrained equations, as well as nonholonomic Chetaev and non-Chetaev type constrained condition equations et al. In chapter three, we look into the Noether symmetries and conserved quantities for the discrete constrained systems. The criterion equations, the discrete constrained restricted equations, and the condition equations of obtaining Noether conserved quantities et al. are deduced for discrete Lagrangian system, discrete Hamiltonian system, discrete non-conservative Lagrangian and Hamiltonian systems, discrete system with variable mass, discrete system with dependent variables, discrete nonholonomic systems with Chetaev also non-Chetaev type constrains, discrete system with unilateral constraints. In chapter four, we make a study of the Mei symmetries and corresponding conserved quantities of the discrete constrained systems. The determining equations and the discrete restricted equations of the Mei symmetries, and the criterion equations of obtaining the Mei conserved quantities are derived for discrete Lagrangian system, discrete Hamiltonian system, discrete non-conservative Lagrangian and Hamiltonian systems, discrete system with variable mass, discrete system with dependent variables, discrete nonholonomic systems with Chetaev also non-Chetaev type constrains, discrete system with unilateral constraints. In chapter five, the Lie symmetries and conserved quantities of the discrete constrained systems are researched. Moreover, the determining equations and the constrained restricted equations of the Lie symmetries, the condition equations of obtaining the Noether also Mei conserved quantities from Lie symmetries et al. are discussed for discrete Lagrangian system, discrete Hamiltonian system, discrete non-conservative Lagrangian and Hamiltonian systems, discrete system with variable mass, discrete system with dependent variables, discrete nonholonomic systems with Chetaev also non-Chetaev type constrains, discrete system with unilateral constraints. In chapter six, we analyze several interrelated symmetries and corresponding conserved quantities. The relationships between Noether symmetry, Mei symmetry and Lie symmetry are clarified. Furthermore, the criterion equations of Noether-Lie symmetry, Lie-Mei symmetry, Noether-Mei symmetry, and the condition equations of acquiring conserved quantities from these symmetries for discrete Lagrangian system and discrete Hamiltonian system are presented. In chapter seven, we summarize the main results of our research and envision the future research directions.