Mei Symmetry and New Conserved Quantity of Tzenoff Equations for Holonomic Systems

A new conserved quantity is deduced from Mei symmetry of Tzenoff equations for holonomic systems. The expression of this new conserved quantity is given, and the determining equation to induce this new conserved quantity is presented. The results exhibit that this new method is easier to find more conserved quantities than the previously reported ones. Finally, application of this new result is presented by a practical example.     

Mei Symmetry and Mei Conserved Quantity for a Non-holonomic System of Non-Chetaev＇s Type with Unilateral Constraints in Nielsen Style

Mei symmetry and Mei conserved quantity for a non-holonomic system of non-Chetaev＇s type with unilateral constraints in the Nielsen style are studied. The differential equations of motion for the system above are established. The definition and the criteria of Mei symmetry, conditions, and expressions of Mei conserved quantity deduced directly from the Mei symmetry are given. An example is given to illustrate the application of the results.     

Mei Symmetry and Conserved Quantity of Tzenoff Equations for Nonholonomic Systems of Non-Chetaev＇s Type

Mei symmetry of Tzenoff equations for nonholonomic systems of non-Chetaev＇s type under the infinitesimal transformations of groups is studied. Its definitions and discriminant equations of Mei symmetry are given. Sufficient and necessary condition of Lie symmetry deduced by the Mei symmetry is also given. Hojman conserved quantity of Tzenoff equations for the systems through Lie symmetry in the condition of special Mei symmetry is obtained.     

Lie Symmetries and Non-Noether Conserved Quantities of Nonholonomic Systems

A new conservation theorem of the nonholonomic systems is studied. The conserved quantity is only constructed in terms of a general Lie group of transformation vector of the dynamical equations. Firstly, we establish the dynamical equations of thc nonholonomic systcms and the determining equations of Lie symmetry. Next, the theorem of non-Noether conserved quantity is deduced. Finally, we give an example to illustrate the application of the result.     

Weakly Noether Symmetry for Nonholonomic Systems of Chetaev＇s Type

In the paper [J. of Beijing Institute of Technology 26 （2006） 285] the authors provided the definition of weakly Noether symmetry. We now discuss the weakly Noether symmetry for non-holonomic system of Chetaev＇s type, and present expressions of three kinds of conserved quantities by weakly Noether symmetry. Finally, the application of this new result is shown by a practical example.     

非完整系统Tzénoff方程的Mei对称性和守恒量

研究了在群的无限小变换下非完整系统Tzénoff方程的Mei对称性，给出了非完整系统Tzénoff方程Mei对称性的定义和判据方程.给出了这种Mei对称性导致Lie对称性的充要条件，通过特殊Mei对称性条件下的Lie对称性，找到了非完整系统Tzénoff方程的Hojman守恒量.     

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.     

Symmetry and Hojman Conserved Quantity of Tzénoff Equations for Unilateral Holonomic System

Symmetry of Tzénoff equations for unilateral holonomic system under the infinitesimal transformations of groups is investigated. Its definitions and discriminant equations of Mei symmetry and Lie symmetry of Tzénoff equations are given. Sufficient and necessary condition of Lie symmetry deduced by the Mei symmetry is also given. Hojman conserved quantity of Tzénoff equations for the system above through special Lie symmetry and Lie symmetry in the condition of special Mei symmetry respectively is obtained.     

Symmetry and Hojman Conserved Quantity of Tzénoff Equations for Unilateral Holonomic System

Symmetry of Tzénoff equations for unilateral holonomic system under the infinitesimal transformations equations are given. Sufficient and necessary condition of Lie symmetry deduced by the Mei symmetry is also given.in the condition of special Mei symmetry respectively is obtained.     

Structure equation and Mei conserved quantity for Mei symmetry of Appell equation

This paper investigates structure equation and Mei conserved quantity of Mei symmetry of Appell equations for non-Chetaev nonholonomic systems. Appell equations and differential equations of motion for non-Chetaev nonholonomic mechanical systems are established. A new expression of the total derivative of the function with respect to time t along the trajectory of a curve of the system is obtained, the definition and the criterion of Mei symmetry of Appell equations under the infinitesimal transformations of groups are also given. The expressions of the structure equation and the Mei conserved quantity of Mei symmetry in the Appell function are obtained. An example is given to illustrate the application of the results.     

Special Lie-Mei Symmetry and Conserved Quantities of Appell Equations Expressed by Appell Function

Special Lie-Mei symmetry and conserved quantities for Appell equations expressed by Appell functions in a holonomic mechanical system are investigated. On the basis of the Appell equation in a holonomic system, the definition and the criterion of special Lie-Mei symmetry of Appell equations expressed by Appell functions are given. The expressions of the determining equation of special Lie-Mei symmetry of Appell equations expressed by Appell functions, Hojman conserved quantity and Mei conserved quantity deduced from special Lie-Mei symmetry in a holonomic mechanical system are gained. An example is given to illustrate the application of the results.     

A New Type of Conserved Quantity of Mei Symmetry for Holonomic Mechanical System

A new type of conserved quantity which is directly induced by the Mei symmetry of the holonomic system is studied. Firstly, the definition and criterion of the Mei symmetry for a holonomic mechanical system is given. Secondly, the condition of existence of the new conserved quantity as well as its form is obtained. Lastly, an example is given to illustrate the application of the results.     

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.     

变质量非完整系统Tzénoff方程的Lie对称性与其导出的守恒量

航天器运行系统大都属于变质量力学系统,变质量力学系统的对称性和守恒量隐含着航天系统更深刻的物理规律.本文首先导出了变质量非完整力学系统的Tzénoff方程,然后研究了变质量非完整力学系统Tzénoff方程的Lie对称性及其所导出的守恒量,给出了这种守恒量的函数表达式和导出这种守恒量的判据方程.该研究结果对进一步探究变质量系统所遵循的守恒规律具有一定的理论价值.     

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.     

Rosenberg问题的对称性与守恒量

Rosenberg问题是一个典型而不太复杂的非完整系统问题.本文利用非完整系统的Noether对称性理论来求这个非完整力学问题的守恒量,进而得到问题的最终解.     

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     

Two New Types of Conserved Quantities Deduced from Noether Symmetry for Nonholonomic Mechanical System

For a nonholonomic mechanical system, the generalized Mei conserved quantity and the new generalized Hojman conserved quantity deduced from Noether symmetry of the system are studied. The criterion equation of the Noether symmetry for the system is got. The conditions under which the Noether symmetry can lead to the two new conserved quantities are presented and the forms of the conserved quantities are obtained. Finally, an example is given to illustrate the application of the results.