A New Method of Fractional Dynamics,i.e., Fractional Mei Symmetrical Method for Finding Conserved Quantity,and its Applications to Physics

In this paper, we present the fractional Mei symmetrical method of finding conserved quantity and explore its applications to physics. For the fractional generalized Hamiltonian system, we introduce the fractional infinitesimal transformation of Lie groups and, under the transformation, give the fractional Mei symmetrical definition, criterion and determining equation. Then, we present the fractional Mei symmetrical theorem of finding conserved quantity. As the fractional Mei symmetrical method’s applications, we respectively find the conserved quantities of a fractional general relativistic Buchduhl model, a fractional three-body model and a fractional Robbins–Lorenz model.     

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 Birkhoff systems under second-class Mei symmetry

This paper proposes a new concept of the conformal invariance and the conserved quantities for Birkhoff systems under second-class Mei symmetry.The definition about conformal invariance of Birkhoff systems under second-class Mei symmetry is given.The conformal factor in the determining equations is found.The relationship between Birkhoff system’s conformal invariance and second-class Mei symmetry are discussed.The necessary and sufficient conditions of conformal invariance,which are simultaneously of second-class symmetry,are given.And Birkhoff system’s conformal invariance may lead to corresponding Mei conserved quantities,which is deduced directly from the second-class Mei symmetry when the conformal invariance satisfies some conditions.Lastly,an example is provided to illustrate the application of the result.     

完整系统Appell方程Mei对称性的共形不变性与守恒量

研究完整系统Appell方程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, 定义完整系统动力学方程的Mei对称性和共形不变性, 给出该系统Mei对称性共形不变性的确定方程. 利用规范函数满足的结构方程导出系统相应的Mei守恒量. 举例说明结果的应用. 关键词： Appell方程 Mei对称性 共形不变性 Mei守恒量     

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     

广义Hamilton系统的Mei对称性导致的Mei守恒量

研究广义Hamilton系统的Mei对称性导致的守恒量. 首先,在群的一般无限小变换下给出广义Hamilton系统的Mei对称性的定义、判据和确定方程;其次,研究系统的Mei守恒量存在的条件和形式,得到Mei对称性直接导致的Mei守恒量; 而后,进一步给出带附加项的广义Hamilton系统Mei守恒量的存在定理; 最后,研究一类新的三维广义Hamilton系统,并研究三体问题中3个涡旋的平面运动. 关键词： 广义Hamilton系统 Mei对称性 Mei守恒量 三体问题     

相对运动完整系统Appell方程Mei对称性的共形不变性与守恒量

研究相对运动完整系统Appell方程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, 给出相对运动完整系统Appell方程的Mei对称性和共形不变性的定义, 导出系统Mei对称性的共形不变性确定方程, 重点讨论系统共形不变性和Mei对称性的关系, 然后借助规范函数满足的结构方程导出系统Mei对称性导致的Mei守恒量表达式, 最后举例说明结果的应用.     

变质量Chetaev型非完整系统Appell方程Mei对称性的共形不变性与守恒量

研究了变质量Chetaev型非完整系统Appell方程Mei对称性的共形不变性和守恒量.在群的无限小变换下,定义了变质量Chetaev型非完整系统Appell方程Mei对称性和共形不变性,给出了该系统Mei对称性的共形不变性确定方程,并推导出系统相应的守恒量表达式.最后,给出了应用算例.     

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.     

一般完整系统Mei对称性的逆问题

动力学逆问题是星际航行学、火箭动力学、规划运动学理论的基本问题. Mei对称性是力学系统的动力学函数在群的无限小变换下仍然满足系统原来的运动微分方程的一种新的不变性. 本文研究广义坐标下一般完整系统的Mei对称性以及与Mei对称性相关的动力学逆问题. 首先, 给出系统动力学正问题的提法和解法. 引入时间和广义坐标的无限小单参数变换群, 得到无限小生成元向量及其一次扩展. 讨论由n个广义坐标确定的一般完整力学系统的运动微分方程, 将其Lagrange函数和非势广义力作无限小变换, 给出系统运动微分方程的Mei对称性定义, 在忽略无限小变换的高阶小量的情况下得到Mei对称性的确定方程, 借助规范函数满足的结构方程导出系统Mei对称性导致的Noether守恒量. 其次, 研究系统Mei对称性的逆问题. Mei对称性的逆问题的提法是: 由已知守恒量来求相应的Mei对称性. 采取的方法是将已知积分当作由Mei对称性导致的Noether守恒量, 由Noether逆定理得到无限小变换的生成元, 再由确定方程来判断所得生成元是否为Mei对称性的. 然后, 讨论生成元变化对各种对称性的影响. 结果表明, 生成元变化对Noether和Lie对称性没有影响, 对Mei 对称性有影响, 但在调整规范函数时, 若满足一定条件, 生成元变化对Mei对称性也可以没有影响. 最后, 举例说明结果的应用.     

Hamilton系统的Mei对称性、Noether对称性和Lie对称性

研究Hamilton系统的形式不变性即Mei对称性，给出其定义和确定方程.研究Hamilton系统的Mei对称性与Noether对称性、Lie对称性之间的关系，寻求系统的守恒量.给出一个例子说明本文结果的应用. 关键词： Hamilton系统 Mei对称性 Noether对称性 Lie对称性 守恒量     

Conformal invariance and generalized Hojman conserved quantities of mechanico-electrical systems

This paper studies conformal invariance and generalized Hojman conserved quantities of mechanico-electrical systems. The definition and the determining equation of conformal invariance for mechanico-electrical systems are provided. The conformal factor expression is deduced from conformal invariance and Lie symmetry under the infinitesimal single-parameter transformation group. The generalized Hojman conserved quantities from the conformal invariance of the system are given. An example is given to illustrate the application of the result.     

Conformal Invariance and Conserved Quantities of General Holonomic Systems

Conformed invariance and conserved quantities of general holonomic systems are studied. A one-parameter infinitesimal transformation group and its infinitesimal transformation vector of generators are described. The definition of conformal invariance and determining equation for the system are provided. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The necessary and suttlcient condition, that conformal invariance of the system would be Lie symmetry, is obtained under the infinitesimal one-parameter transformation group. The corresponding conserved quantity is derived with the aid of a structure equation. Lastly, an example is given to demonstrate the application of the result.     

完整系统Tzénoff方程的Mei对称性直接导致的另一种守恒量

研究了完整力学系统Tzénoff方程Mei对称性直接导致的另一种守恒量,给出了这种守恒量的函数表达式和导致这种守恒量的确定方程.利用该方法比以往更易找到守恒量.最后举例说明了新结果的应用.     

广义Hamilton系统的共形不变性与Hojman守恒量

研究了广义Hamilton系统在无限小变换下的共形不变性,推导出共形不变性的确定方程,找到在无限小变换下的共形不变性并且是Lie对称性的共形因子,最后导出广义Hamilton系统的运动微分方程共形不变时的Hojman守恒量,并给出应用算例. 关键词： 广义Hamilton系统 共形不变性 Hojman守恒量 确定方程     

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.     

New non-Noether conserved quantities of mechanical system in phase space

This paper focuses on studying non-Noether conserved quantities of Lie symmetry and of form invariance for a mechanical system in phase space under the general infinitesimal transformation of groups. We obtain a new non-Noether conserved quantity of Lie symmetry of the system, and Hojman and Mei's results are of special cases of our conclusion. We find a condition under which the form invariance of the system will lead to a Lie symmetry, and, further, obtain a new non-Noether conserved quantity of form invariance of the system. An example is given finally to illustrate these results.     

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.     

Hamilton系统Mei对称性的一种新守恒量

研究Hamilton系统的Mei对称性直接导致的一种新守恒量. 给出Hamilton系统的Mei对称性的定义和判据方程，引入谐调函数，得到系统Mei对称性直接导致新守恒量的条件和形式，并给出应用算例. 结果表明， 谐调函数可根据寻找规范函数的需要适当选取, 从而使规范函数的寻求变得比较容易，而且由于谐调函数的选取具有多样性, 因此能够找到系统Mei对称性的更多的守恒量. 关键词： Hamilton系统 Mei对称性 新守恒量     

Conformal invariance and integration of first-order differential equations

This paper studies a conformal invariance and an integration of first-order differential equations. It obtains the corresponding infinitesimal generators of conformal invariance by using the symmetry of the differential equations, and expresses the differential equations by the equations of a Birkhoff system or a generalized Birkhoff system. If the infinitesimal generators are those of a Noether symmetry, the conserved quantity can be obtained by using the Noether theory of the Birkhoff system or the generalized Birkhoff system.