Noether——Lie symmetry and conserved quantities of mechanical system in phase space

In this paper, a new kind of symmetry and its conserved quantities of a mechanical system in phase space are studied. The definition of this new symmetry, i.e. a Noether--Lie symmetry, is presented, and the criterion of this symmetry is also given. The Noether conserved quantity and the generalized Hojman conserved quantity of the Noether--Lie symmetry of the system are obtained. The Noether--Lie symmetry contains the Noether symmetry and the Lie symmetry, and has more generalized significance.     

Unified Symmetry of Nonholonomic Mechanical Systems of Non-Chetaev‘s Type with Unilateral Constraints

The definition and the criterion for a unified symmetry of nonholonomic mechanical systems of nonChetaev‘s type with unilateral constraints are presented based on the total time derivative along the trajectory of the system. A new conserved quantity, as well as the Noether conserved quantity and the Hojman conserved quantity,jeduced from the unified symmetry, is obtained. An example is given to illustrate the application of the results.     

Unified Symmetry of Nonholonomic Mechanical Systems of Non-Chetaev＇s Type with Unilateral Constraints

The definition and the criterion for a unified symmetry of nonholonomic mechanical systems of non- Chetaev＇s type with unilateral constraints are presented based on the total time derivative along the trajectory of the system. A new conserved quantity, as well as the Noether conserved quantity and the Hojman conserved quantity, deduced froth the unified symmetry, is obtained. An example is given to illustrate the application of the results.     

Mei symmetry and Mei conserved quantity of Nielsen equations for a non-holonomic system of Chetaev’s type with variable mass

Mei symmetry and Mei conserved quantity of Nielsen equations for a non-holonomic, non-conservative system of Chetaev's type with variable mass are studied. The differential equations of motion of the Nielsen equation for the system, the definition and criterion of Mei symmetry, and the condition and the form of Mei conserved quantity deduced directly by Mei symmetry for the system are obtained. An example is given to illustrate the application of the results.     

Unified Symmetry of Nonholonomic Mechanical Systems with Non-Chetaev‘s Type Constraints

Based on the total time derivative along the trajectory of the system, the unified symmetry of nonholonomic mechanical system with non-Chetaev‘s type constraints is studied. The definition and criterion of the unified symmetry of nonholonomic mechanical systems with non-Chetaev‘s type constraints are given. A new conserved quantity, as well as the Noether conserved quantity and the Hojman conserved quantity, deduced from the unified symmetry, is obtained.Two examples are given to illustrate the application of the results.     

Unified Symmetry of Nonholonomic System of Non-Chetaev＇s Type with Variable Mass in Event Space

The unified symmetry of a nonholonomic system of non-Chetaev＇s type with variable mass in event space is studied. The differential equations of motion of the system are given. Then the definition and the criterion of the unified symmetry for the system are obtained. Finally, the Noether conserved quantity, the Hojman conserved quantity, and a new type of conserved quantity are deduced from the unified symmetry of the nonholonomic system of non-Chetaev＇s type with variable mass in event space at one time. An example is given to illustrate the application of the results.     

Unified Symmetry of Nonholonomic Mechanical Systems with Non-Chetaev＇s Type Constraints

Based on the total time derivative along the trajectory of the system, the unified symmetry of nonholonomic mechanical system with non-Chetaev＇s type constraints is studied. The definition and criterion of the unified symmetry of nonholonomic mechanical systems with non-Ohetaev＇s type constraints are given. A new conserved quantity, as well as the Noether conserved quantity and the Hojman conserved quantity, deduced from the unified symmetry, is obtained. Two examples are given to illustrate the application of the results.     

Mei Symmetry and Hojman Conserved Quantity of Nonholonomic Controllable Mechanical System

A non-Noether conserved quantity, i.e., Hojman conserved quantity, constructed by using Mei symmetry for the nonholonomic controllable mechanical system, is presented. Under general infinitesimal transformations, the determining equations of the special Mei symmetry, the constrained restriction equations, the additional restriction equations, and the definitions of the weak Mei symmetry and the strong Mei symmetry of the nonholonomic controllable mechanical system are given. The condition under which Mei symmetry is a Lie symmetry is obtained. The form of the Hojman conserved quantity of the corresponding holonomic mechanical system, the weak Hojman conserved quantity and the strong Hojman conserved quantity of the nonholonomie controllable mechanical system are obtained. An example is given to illustrate the application of the results.     

Noether symmetry and Lie symmetry of discrete holonomic systems with dependent coordinates

The Noether symmetry, the Lie symmetry and the conserved quantity of discrete holonomic systems with dependent coordinates are investigated in this paper. The Noether symmetry provides a discrete Noether identity and a conserved quantity of the system. The invariance of discrete motion equations under infinitesimal transformation groups is defined as the Lie symmetry, and the condition of obtaining the Noether conserved quantity from the Lie symmetry is also presented. An example is discussed to show the applications of the results.     

Lie symmetry and its generation of conserved quantity of Appell equation in a dynamical system of the relative motion with Chetaev-type nonholonomic constraints

Lie symmetry and conserved quantity deduced from Lie symmetry of Appell equations in a dynamical system of relative motion with Chetaev-type nonholonomic constraints are studied.The differential equations of motion of the Appell equation for the system,the definition and criterion of Lie symmetry,the condition and the expression of generalized Hojman conserved quantity deduced from Lie symmetry for the system are obtained.The condition and the expression of Hojman conserved quantity deduced from special Lie symmetry for the system under invariable time are further obtained.An example is given to illustrate the application of the results.     

规范变换对Birkhoff系统对称性的影响

研究Birkhoff系统规范变换对其Noether对称性、Lie对称性和Mei对称性的影响.在一定条件下,Noether对称性和守恒量不改变.Lie对称性和Hojaman守恒量仍保持不变.Mei对称性和新型守恒量可能变化,得到了Mei对称性和新型守恒量保持不变的条件.举例说明结果的应用. 关键词： Birkhoff系统 规范变换 对称性 守恒量     

一类动力学方程的Mei对称性

研究一类动力学方程的Mei对称性的定义和判据，由Mei对称性通过Noether对称性可找到Noether守恒量.由Mei对称性通过Lie对称性可找到Hojman守恒量.同时，也可找到一类新型守恒量.     

Poincaré-Chetaev方程的统一对称性

研究Poincaré-Chetaev方程的统一对称性及其导致的守恒量. 给出Poincaré-Chetaev方程统一对称性的定义和判据, 得到Poincaré-Chetaev方程统一对称性的三种守恒量: Noether守恒量、Hojman守恒量和Mei守恒量. 关键词： Poincaré-Chetaev方程 统一对称性 守恒量     

非完整系统的Noether对称性与Hojman守恒量

研究非完整力学系统的Noether对称性导致的非Noether守恒量——Hojman守恒量. 在时间不变的特殊无限小变换下，给出系统的特殊Noether对称性与守恒量，并给出特殊Noether对称性导致特殊Lie对称性的条件. 由系统的特殊Noether对称性，得到相应完整系统的Hojman守恒量以及非完整系统的弱Hojman守恒量和强Hojman守恒量. 给出一个例子说明本结果的应用 关键词： 分析力学 非完整系统 Noether对称性 非Noether守恒量 Hojman守恒量     

Mei Symmetry and Hojman Conserved Quantity of Nonholonomic Controllable Mechanical System

A non-Noether conserved quantity, i.e., Hojman conserved quantity, constructed by using Mei symmetry for the nonholonomic controllable mechanical system, is presented. Under general infinitesimal transformations, the determining equations of the special Mei symmetry, the constrained restriction equations, the additional restriction equations, and the definitions of the weak Mei symmetry and the strong Mei symmetry of the nonholonomic controllable mechanical system are given. The condition under which Mei symmetry is a Lie symmetry is obtained. The form of the Hojman conserved quantity of the corresponding holonomic mechanical system, the weak Hojman conserved quantity and the strong Hojman conserved quantity of the nonholonomic controllable mechanical system are obtained. An example is given to illustrate the application of the results.     

非完整力学系统的Noether-Lie对称性

研究了非完整力学系统的一种新对称性——Noether-Lie对称性及其守恒量. 给出了非完整力学系统Noether -Lie对称性的定义和判据，提出系统的Noether-Lie对称性导致Noether守恒量和广义Hojman守恒量的定理. 举例说明了结果的应用. Hojman守恒量是所给出的广义Hojman守恒量的特例. 关键词： 非完整力学系统 Noether-Lie对称性 Noether守恒量 广义Hojman守恒量     

完整系统Nielsen方程的统一对称性与守恒量

研究完整系统Nielsen方程的统一对称性与守恒量.在完整系统Nielsen方程的基础上,首先给出了Nielsen方程的Noether对称性、Lie对称性和Mei对称性与守恒量,其次给出了Nielsen方程的统一对称性的定义和判据,得到Nielsen方程的统一对称性导致的Noether守恒量、Hojman守恒量和Mei守恒量.举例说明结果的应用.     

Unified Symmetry of Hamilton Systems

The definition and the criterion of a unified symmetry for a Hamilton system are presented. The sufficient condition under which the Noether symmetry is a unified symmetry for the system is given. A new conserved quantity,as well as the Noether conserved quantity and the Hojman conserved quantity, deduced from the unified symmetry, is obtained. An example is finally given to illustrate the application of the results.     

Unified Symmetry of Hamilton Systems

The definition and the criterion of a unified symmetry for a Hamilton system are presented. The sufficient condition under which the Noether symmetry is a unified symmetry for the system is given. A new conserved quantity, as well as the Noether conserved quantity and the Hojman conserved quantity deduced from the unified symmetry, is obtained. An example is finally given to illustrate the application of the results.     

A New Type of Conserved Quantity of Mei Symmetry for Lagrange Systems

A new type of conserved quantity, which is induced from the Mei symmetry of Lagrange systems, is studied. The conditions for that the new type of conserved quantity exists and the form of the new type of conserved quantity are obtained. An illustrated example is given. The Noether conserved quantity induced from the Mei symmetry of Lagrange systems is a special case of the new type of conserved quantity given in this paper.