Conformal invariance and Noether symmetry, Lie symmetry of holonomic mechanical systems in event space

This paper is devoted to studying the conformal invariance and Noether symmetry and Lie symmetry of a holonomic mechanical system in event space. The definition of the conformal invariance and the corresponding conformal factors of the holonomic system in event space are given. By investigating the relation between the conformal invariance and the Noether symmetry and the Lie symmetry, expressions of conformal factors of the system under these circumstances are obtained, and the Noether conserved quantity and the Hojman conserved quantity directly derived from the conformal invariance are given. Two examples are given to illustrate the application of the results.     

变质量完整系统的共形不变性和Noether对称性及Lie对称性

研究变质量完整系统在无限小变换下的共形不变性与Noether对称性和Lie对称性.首先,给出了变质量完整系统的共形不变性的定义;其次,研究了系统的共形不变性与Noether对称性之间的关系,得到了共形不变性导致的Noether守恒量;最后,研究了系统的共形不变性与Lie对称性之间的关系,得到了共形不变性同时是Lie对称性导致的Hojman守恒量.最后举例说明了结果的应用.     

单面Chetaev型非完整系统的共形不变性、Noether对称性和Lie对称性

研究单面Chetaev型非完整系统在无限小变换下的共形不变性及其与Noether对称性和Lie对称性的关系. 首先,给出了单面Chetaev型非完整系统的共形不变性的定义; 其次,研究了系统的共形不变性与Noether对称性之间的关系;最后, 研究了系统的共形不变性与Lie对称性之间的关系,得到了共形不变性同时是Lie 对称性导致的Hojman守恒量.最后分别举例说明了结果的应用.     

Conformal Invariance and Noether Symmetry,Lie Symmetry of Birkhoffian Systems in Event Space

This paper focuses on studying a conformal invariance and a Noether symmetry, a Lie symmetry for a Birkhoffian system in event space. The definitions of the conformal invariance of the system are given. By investigation on the relations between the conformal invariance and the Noether symmetry, the conformal invariance and the Lie symmetry, the expressions of conformal factors of the system under these circumstances are obtained. The Noether conserved quantities and the Hojman conserved quantities directly derived from the conformal invariance are given. Two examples are given to illustrate the application of the results.     

一般完整系统Mei对称性的共形不变性与守恒量

研究一般完整系统Mei对称性的共邢不变性与守恒量.引入无限小单参数变换群及其生成元向量,定义一般完整系统动力学方程的Mei对称性共形不变性,借助Euler算子导出Mei对称性共形不变性的相关条件,给出其确定方程.讨论共形不变性与Noether对称性、Lie对称性以及Mei对称性之间的关系.利用规范函数满足的结构方程得到系统相应的守恒量.举例说明结果的应用. 关键词： 一般完整系统 Mei对称性 共形不变性 守恒量     

完整系统Appell方程Lie对称性的共形不变性与Hojman守恒量

研究完整系统Appell方程Lie对称性的共形不变性与Hojman守恒量.在时间不变的特殊无限小变换下,定义完整系统动力学方程的Lie对称性和共形不变性,给出该系统Lie对称性共形不变性的确定方程及系统的Hojman守恒量,并举例说明结果的应用.     

离散差分变分Hamilton系统的Lie对称性与Noether守恒量

研究离散差分Hamilton系统的Lie对称性与Noether守恒量. 根据扩展的时间离散力学变分原理构建Hamilton系统的差分动力学方程.定义离散系统运动差分方程在无限小变换群下的不变性为Lie对称性, 导出由Lie对称性得到系统离散Noether守恒量的判据. 举例说明结果的应用. 关键词： 离散力学 差分Hamilton系统 Lie对称性 Noether守恒量     

离散差分序列变质量Hamilton系统的Lie对称性与Noether守恒量

本文研究离散差分序列变质量Hamilton系统的Lie对称性与Noether守恒量. 构建了离散差分序列变质量Hamilton系统的差分动力学方程, 给出了离散差分序列变质量Hamilton系统差分动力学方程在无限小变 换群下的Lie对称性的确定方程和定义, 得到了离散力学系统Lie对称性导致Noether守恒量的条件及形式, 举例说明结果的应用. 关键词： 离散力学 Hamilton系统 Lie对称性 Noether守恒量     

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.     

Conformal invariance and conserved quantity of Hamilton systems

This paper studies conformal invariance and conserved quantities of Hamilton system. The definition and the determining equation of conformal invariance for Hamilton system are provided. The relationship between the conformal invariance and the Lie symmetry are discussed, and the necessary and sufficient condition that the conformal invariance would be the Lie symmetry of the system under the infinitesimal one-parameter transformation group is deduced. It gives the conserved quantities of the system and an example for illustration.     

广义Hamilton系统的共形不变性与Mei守恒量

研究广义Hamilton系统在无限小变换下的共形不变性与Mei对称性,给出系统共形不变性同时是Mei对称性的充分必要条件,得到广义Hamilton系统共形不变性导致的Mei守恒量,举例说明结果的应用.     

Conformal invariance and conserved quantities of general holonomic systems in phase space

This paper studies the conformal invariance and conserved quantities of general holonomic systems in phase space. The definition and the determining equation of conformal invariance for general holonomic systems in phase space are provided. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The relationship between the conformal invariance and the Lie symmetry is discussed, and the necessary and sufficient condition that the conformal invariance would be the Lie symmetry of the system under the infinitesimal single-parameter transformation group is deduced. The conserved quantities of the system are given. An example is given to illustrate the application of the result.     

Conformal invariance and conserved quantities of a general holonomic system with variable mass

Conformal invariance and conserved quantities of a general holonomic system with variable mass are studied. The definition and the determining equation of conformal invariance for a general holonomic system with variable mass are provided. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The relationship between the conformal invariance and the Lie symmetry is discussed, and the necessary and sufficient condition under which the conformal invariance would be the Lie symmetry of the system under an infinitesimal one-parameter transformation group is deduced. The conserved quantities of the system are given. An example is given to illustrate the application of the result.     

Conformal invariance and conserved quantity of third-order Lagrange equations for non-conserved mechanical systems

This paper studies conformal invariance and conserved quantity of third-order Lagrange equations for non-conserved mechanical systems. Third-order Lagrange equations, the definition and a determining equation of conformal invariance of the system are presented. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The necessary and sufficient condition that conformal invariance of the system would have Lie symmetry under single-parameter infinitesimal transformations is obtained. The corresponding conserved quantity of conformal invariance is derived with the aid of a structure equation. Lastly, an example is given to illustrate the application of the results.     

Noether symmetry and conserved quantity for a Hamilton system with time delay

In this paper, the Noether symmetries and the conserved quantities for a Hamilton system with time delay are discussed. Firstly, the variational principles with time delay for the Hamilton system are given, and the Hamilton canonical equations with time delay are established. Secondly, according to the invariance of the function under the infinitesimal transformations of the group, the basic formulas for the variational of the Hamilton action with time delay are discussed,the definitions and the criteria of the Noether symmetric transformations and quasi-symmetric transformations with time delay are obtained, and the relationship between the Noether symmetry and the conserved quantity with time delay is studied. In addition, examples are given to illustrate the application of the results.     

Conformal invariance and conserved quantities of Appell systems under second-class Mei symmetry

In this paper we introduce the new concept of the conformal invariance and the conserved quantities for Appell systems under second-class Mei symmetry. The one-parameter infinitesimal transformation group and infinitesimal transformation vector of generator are described in detail. The conformal factor in the determining equations under second-class Mei symmetry is found. The relationship between Appell system’s conformal invariance and Mei symmetry are discussed. And Appell system’s conformal invariance under second-class Mei symmetry may lead to corresponding Hojman conserved quantities when the conformal invariance satisfies some conditions. Lastly, an example is provided to illustrate the application of the result.     

非完整系统的共形不变性导致的新型守恒量

研究了非完整系统的共形不变性与新型守恒量.提出了该系统共形不变性的概念;得出了非完整系统的运动微分方程具有共形不变性并且是Lie对称性的充要条件.利用规范函数满足的新型结构方程,导出系统相应的新型守恒量.最后给出应用算例.     

事件空间中完整系统的Hojman守恒量

研究事件空间中完整力学系统由特殊Lie对称性、Noether对称性和形式不变性导致的Hojman守恒量.列出系统的运动微分方程.给出Lie对称性、Noether对称性和形式不变性的判据，以及三种对称性之间的关系.将Hojman定理推广并应用于事件空间完整系统，得到非Noether守恒量.举例说明结果的应用. 关键词： 分析力学 完整系统 事件空间 对称性 Hojman守恒量     

Chetaev型非完整系统Mei对称性的共形不变性与守恒量

研究Chetaev型非完整系统Mei对称性的共形不变性与守恒量.引入无限小单参数变换群及其生成元向量,给出与Chetaev型非完整系统相应的完整系统的Mei对称性共形不变性定义和确定方程.讨论系统共形不变性与Mei对称性的关系.利用限制方程和附加限制方程得到非完整系统弱Mei对称性和强Mei对称性的共形不变性.借助规范函数满足的结构方程导出系统相应的守恒量,并举例说明结果的应用.     

变质量完整动力学系统的共形不变性与守恒量

研究变质量完整力学系统在无限小变换下微分方程的共形不变性.提出了该系统共形不变性的概念,推导出变质量完整力学系统运动微分方程具有共形不变性,同时又是Lie对称性的充分必要条件.得到由共形不变性导致的Noether守恒量. 关键词： 变质量系统 无限小变换 共形不变 守恒量