离散约束动力学系统的对称性质与守恒量研究
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摘要
运用无限小Lie变换群方法研究离散约束动力学系统的对称性质,利用对称性分析方法寻求系统的离散守恒量。第一章回顾约束力学系统对称性与守恒量的研究概况,给出对称性的普适定义,概述连续和离散约束系统对称性与守恒量研究的意义、方法、历史发展与现状,包括Noether对称性、Mei对称性、Lie对称性和几类联合对称性。第二章研究离散约束系统的动力学方程,给出包含时间变分的全变分原理,建立离散Lagrange系统、离散Hamilton系统、非保守Lagrange与Hamilton系统、离散变质量系统、非独立变量离散系统、非完整Chetaev型与非Chetaev型离散系统、单面约束离散系统的动力学方程与约束方程,包括离散Euler-Lagrange方程、离散正则方程、离散能量演化方程、完整与非完整的离散约束方程、非完整Chetaev型与非Chetaev型的离散约束条件方程等。第三章研究离散约束系统的Noether对称性与守恒量,给出离散Lagrange系统、离散Hamilton系统、非保守Lagrange与Hamilton系统、离散变质量系统、非独立变量离散系统、非完整Chetaev型与非Chetaev型离散系统、单面约束离散系统的Noether对称性的判据方程、离散约束限制方程和得到Noether守恒量的条件方程等。第四章研究离散约束系统的Mei对称性与守恒量,给出离散Lagrange系统、离散Hamilton系统、非保守Lagrange与Hamilton系统、离散变质量系统、非独立变量离散系统、非完整Chetaev型与非Chetaev型离散系统、单面约束离散系统的Mei对称性确定方程、Mei对称性离散限制方程和得到Mei守恒量的判据方程等。第五章研究离散约束系统的Lie对称性与守恒量,给出离散Lagrange系统、离散Hamilton系统、非保守Lagrange与Hamilton系统、离散变质量系统、非独立变量离散系统、非完整Chetaev型与非Chetaev型离散系统的Lie对称性确定方程、Lie对称性约束限制方程,Lie对称性得到Noether守恒量、Mei守恒量的条件方程等。第六章研究离散约束系统的几类联合对称性及其守恒量,讨论离散约束系统Noether对称性、Mei对称性、Lie对称性的关系,给出离散Lagrange系统的Noether-Lie对称性、Lie-Mei对称性、Noether-Mei对称性和统一对称性的判据方程。第七章总结研究的主要结果并展望未来研究的若干方向。
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.
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.
引文
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