非完整系统的Lie对称性与Noether对称性

研究非完整系统的Lie对称性与Noether对称性及其间的关系,具体研究了Chetaev型变量质量非完整系统和非Chetaev型非完整系统的Lie对称性与Noether对称性。给出Lie对称性导致Noether对称性及Noether对称性导致Lie对称性的条件。     

非Четаев型非完整系统的Lie对称性与守恒量

研究非Четаев型非完整系统的Ｌｉｅ对称性．首先利用微分方程在无限小变换下的不变性建立Ｌｉｅ对称所满足的确定方程和限制方程，给出结构方程并求出守恒量；其次研究上述问题的逆问题：根据已知积分求相应的Ｌｉｅ对称性；最后举例说明结果的应用．     

非完整动力学逆问题的基本解法

本文绘出非完整动力学逆问题的两种基本提法和解法,包括运动方程的构成以及运动方程的修改     

非保守力学系统的Lie对称性和守恒量

本文讨论非保守力学系统的Ｌｉｅ对称性，给出由Ｌｉｅ对称性得到力学系统守恒量的条件，并给出了说明性的例子，在文章最后，还说明对于非保守系统，它的Ｎｏｅｔｈｅｒ对称性并不一定是其自身的Ｌｉｅ对称性。     

相空间中有二阶线性单面约束的非完整系统的Lie对称性与守恒量

研究相空间中有二阶线性单面约束的非完整系统的Lie对称性与守恒量.首先根据微分方程在无限小变换下的不变性建立Lie对称性所满足的确定方程和限制方程,给出结构方程和守恒量;其次讨论系统的Lie对称性逆问题;最后举一实例说明结果的应用.     

具有可积微分约束的力学系统的Lie对称性

研究具有可积微分约束的力学系统的Ｌｉｅ对称性与守恒量。采用两种方法：一是用不可积微分约束系统的方法;另一是用积分后降阶系统的方法,研究两种方法之间的关系。     

Szebehely问题对非完整系统的一个推广

本文考虑完整系统的一个动力学逆问题，将完整系统的Ｓｚｅｂｅｈｅｌｙ问题，已知一簇带ｎ－１个参数的轨线可一系统的力函数，推广到非完整系统。     

基于分数阶模型的非完整系统的Mei对称性及其守恒量

为了进一步揭示非完整系统的对称性和守恒量之间的内在关系,提出并研究基于分数阶模型的非完整系统的Mei对称性及其守恒量．首先,根据分数阶d’Alembert-Lagrange原理建立基于分数阶模型的非完整系统的动力学方程．其次,根据动力学方程中的动力学函数经无限小变换后仍满足原方程的不变性,建立分数阶模型下非完整系统的Mei对称性定理,给出Mei守恒量．再次,讨论了几个特例：分数阶Hamilton系统、经典非完整系统和受非完整约束的分数阶Lagrange系统的Mei对称性定理．文末举例说明结果的应用．     

变质量完整力学系统的Hojman守恒量

研究完整力学系统由Noether对称性导致的Hojman守恒量．列写系统的运动微分方程；在时间不变的特殊无限小变换下，研究系统的Noether对称性与Lie对称性，给出Noether对称性为Lie对称性的条件；将Hojman定理推广至变质量系统，并举例说明结果的应用．     

非力学系统的对称性与守恒量(I)——分析力学札记之七

将非力学系统的微分方程化成Ｌａｇｒａｎｇｅ方程形式,引进无限小变换,研究微分方程或Ｈａｍｉｌｔｏｎ作用量在无小变换,研究微分方程或Ｈａｍｉｌｔｏｎ作用量在无限小变换下的不变性,进而给出守恒量存在的条件以及守恒量的形式。     

包含伺服约束的非完整系统的对称性摄动与绝热不变量

建立包含伺服约束系统的运动微分方程,基于包含伺服约束系统对称性与不变量理论研究包含伺服约束系统的对称性摄动与绝热不变量 问题,提出了包含伺服约束系统的高阶绝热不变量概念,证明了精确不变量与绝热不变量存在的条件及形式,并举例说明结果的应用。     

On the Noether symmetry and Lie symmetry of mechanical systems

The Noether symmetry is an invariance of Hamilton action under infinitesimal transformations of time and the coordinates. The Lie symmetry is an invariance of the differential equations of motion under the transformations. In this paper, the relation between these two symmetries is proved definitely and firstly for mechanical systems. The results indicate that all the Noether symmetries are Lie symmetries for Lagrangian systems meanwhile a Noether symmetry is a Lie symmetry for the general holonomic or nonholonomic systems provided that some conditions hold. The project supported by the National Natural Science Foundation of China (19972010)     

Lie symmetries and conserved quantities of second-order nonholonomic mechanical system

The Lie symmetries and the conserved quantities of the second-order nonholonomic mechanical system are studied. Firstly, by using the invariance of the differential equation of motion under the infinitesimal transformations, the determining equations and the restriction equations of the Lie symmetries of the system are established, and the structure equation and the conservative quantities of the Lie symmetries are obtained. Secondly , the inverse problems of the Lie symmetries are studied . Finally , an example is given to illustrate the application of the result.     

Lie symmetries and conserved quantities of holonomic variable mass systems

In this paper, the Lie symmetries and the conserved quantities of the holonomic variable mass systems are studied. By using the invariance of the ordinary differential equations under the infinitesimal transformations, the determining equations and the conserved quantities are given. And an example is given to illustrate the application of the result. Foundation item: the National Natural Science Foundation of China (19572038)     

LIE SYMMETRIES AND CONSERVED QUANTITY OF A BIPED ROBOT

I. INTRODUCTION It is well known there are close relationships between the symmetries and conservation laws inmechanical systems. The symmetric principles are among the key issues in mechanics. Two e?ectivemethods of studying the symmetries and conservation laws of mechanical are Noether’s method[1] andLie’s method. The approach to Lie symmetries was reported in the 19th century, but no applicationin mechanics appeared until 1979[2]. In recent years, studies of Lie’s method have be…     

Form invariance and Lie symmetry of variable mass nonholonomic mechanical system

IntroductionThereisacloserelationbetweenthesymmetryandtheconservedquantityinamechanicalsystem .ModernmethodstofindconservedquantityofamechanicalsystemaremainlyNoethersymmetrymethod[1]andLiesymmetrymethod[2 ].NoethersymmetryisaninvarianceoftheHamiltonactionundertheinfinitesimaltransformations.Liesymmetryisaninvarianceofthedifferentialequationsundertheinfinitesimaltransformations.Inthepasttenyears,aseriesofimportresultshavebeenobtainedonthestudyoftheNoethersymmetryandLiesymmetry[3～12 ].Thefo…     

THE LIE SYMMETRIES AND CONSERVED QUANTITIES OF VARIABLE-MASS NONHOLONOMIC SYSTEM OF NON-CHETAEV''''S TYPE IN PHASE SPACE

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｓｔｕｄｙｏｆｓｙｍｍｅｔｒｙａｎｄｃｏｎｓｅｒｖｅｄｑｕａｎｔｉｔｙｏｆｍｅｃｈａｎｉｃａｌｓｙｓｔｅｍｉｓａｎｉｍｐｏｒｔａｎｔｔｏｐｉｃｉｎｍａｔｈｅｍａｔｉｃｓ,ｍｅｃｈａｎｉｃｓａｎｄｐｈｙｓｉｃｓ .ＴｈｅｍｏｄｅｒｎｔｈｅｏｒｉｅｓｏｆｓｙｍｍｅｔｒｙａｎｄｃｏｎｓｅｒｖｅｄｑｕａｎｔｉｔｙｏｆｍｅｃｈａｎｉｃａｌｓｙｓｔｅｍｉｎｃｌｕｄｅＮｏｅｔｈｅｒｓｙｍｍｅｔｒｙｔｈｅｏｒｙａｎｄＬｉｅｓｙｍｍｅｔｒｙｔｈｅｏｒｙ .Ｉｎ 1979Ｍ .Ｌｕｔｚｋｙａｎｄｏ…