A New Type of Non-Noether Adiabatic Invariants for Disturbed Lagrangian Systems： Adiabatic Invariants of Generalized Lutzky Type

For a Lagrangian system with the action of small disturbance, the Lie symmetrical perturbation and a new type of non-Noether adiabatic invariant are presented in general infinitesimal transformation groups. On the basis of the invariance of disturbed Lagrangian systems under general infinitesimal transformations, the determining equations of Lie symmetries of the system are constructed. Based on the definition of higher-order adiabatic invariants of a mechanical system, a new type of adiabatic invariant, i.e. generalized Lutzky adiabatic invariants, of a disturbed Lagrangian system are obtained by investigating the perturbation of Lie symmetries t＇or a Lagrangian system with the action of small disturbance. Finally, an example is given to illustrate the application of the method and results.     

Lie Symmetrical Perturbation and Adiabatic Invariants of Generalized Hojman Type for Relativistic Birkhoffian Systems

For a relativistic Birkhoffian system, the Lie symmetrical perturbation and adiabatic invariants of generalized Bojman type are studied under general infinitesimal transformations. On the basis of the invariance of relativistic Birkhotfian equations under general infinitesimal transformations,Lie symmetrical transformations of the system are constructed, which only depend on the Birkhoffian variables. The exact invariants in the form of generalized Hojman conserved quantities led by the Lie symmetries of relativistic Birkhoffian system without perturbations are given. Based on the definition of higher-order adiabatic invariants of a mechanical system, the perturbation of Lie symmetries for relativistic Birkhoffian system with the action of small disturbance is investigated, and a new type of adiabatic invariants of the system is obtained. In the end of the paper, an example is given to illustrate the application of the results.     

Lie symmetrical perturbation and adiabatic invariants of generalized Hojman type for Lagrange systems

Based on the invariance of differential equations under infinitesimal transformations of group, Lie symmetries, exact invariants, perturbation to the symmetries and adiabatic invariants in form of non-Noether for a Lagrange system are presented. Firstly, the exact invariants of generalized Hojman type led directly by Lie symmetries for a Lagrange system without perturbations are given. Then, on the basis of the concepts of Lie symmetries and higher order adiabatic invariants of a mechanical system, the perturbation of Lie symmetries for the system with the action of small disturbance is investigated, the adiabatic invariants of generalized Hojman type for the system are directly obtained, the conditions for existence of the adiabatic invariants and their forms are proved. Finally an example is presented to illustrate these results.[第一段]     

变质量单面完整约束系统Lie对称性的摄动与广义Hojman型绝热不变量

研究变质量单面完整约束系统Lie对称性的摄动与广义Hojman型绝热不变量.首先通过一般无限小变换下的Lie对称性得到广义Hojman型的守恒量；然后基于力学系统高阶绝热不变量的定义，研究小扰动作用下系统Lie对称性的摄动，得到系统广义Hojman型绝热不变量；最后举例说明结果的应用. 关键词： 变质量 单面完整约束 对称性 摄动 绝热不变量     

Adiabatic invariants of generalized Lutzky type for disturbed holonomic nonconservative systems

Based on the definition of higher-order adiabatic invariants of a mechanical system, a new type of adiabatic invariants, i.e. generalized Lutzky adiabatic invariants, of a disturbed holonomic nonconservative mechanical system are obtained by investigating the perturbation of Lie symmetries for a holonomic nonconservative mechanical system with the action of small disturbance. The adiabatic invariants and the exact invariants of the Lutzky type of some special cases, for example, the Lie point symmetrical transformations, the special Lie symmetrical transformations, and the Lagrange system, are given. And an example is given to illustrate the application of the method and results.     

相空间中离散力学系统对称性的摄动与Hojman型绝热不变量

研究相空间中离散力学系统对称性的摄动与绝热不变量.列出相空间中未受扰离散力学系统的特殊Lie对称性导致的Hojman型精确不变量.基于相空间中力学系统的高阶绝热不变量的定义，研究在小扰动作用下系统Lie对称性的摄动，得到了相空间中离散力学系统的一类新的绝热不变量——Hojman型绝热不变量.举例说明结果的应用. 关键词： 相空间 Lie对称性 摄动 绝热不变量     

Lie Symmetries, Perturbation to Symmetries and Adiabatic Invariants of a Generalized Birkhoff System

We study the perturbation to symmetries and adiabatic invariants of a generalized Birkhoff system. Based on the invariance of differential equations under infinitesimal transformations, Lie symmetries, laws of conservations, perturbation to the symmetries and adiabatic invariants of the generalized Birkhoff system are presented. First, the concepts of Lie symmetries and higher order adiabatic invariants of the generalized Birkhoff system are proposed. Then, the conditions for the existence of the exact invariants and adiabatic invariants are proved, and their forms are given. Finally, an example is presented to illustrate the method and results.     

Lagrange系统对称性的摄动与Hojman型绝热不变量

研究Lagrange系统对称性的摄动与绝热不变量.列出未受扰Lagrange系统的Lie对称性导致的Hojman守恒量；基于力学系统的高阶绝热不变量的定义，研究在小扰动作用下Lagrange系统Lie对称性的摄动，得到了系统的一类Hojman形式的绝热不变量.并举例说明结果的应用. 关键词： Lagrange系统 对称性 摄动 绝热不变量     

Birkhoff系统的一类新型绝热不变量

研究Birkhoff系统对称性的摄动与绝热不变量.给出了未受扰Birkhoff系统的Lie对称性导致的Hojman型精确不变量.基于力学系统的高阶绝热不变量的定义，研究在小扰动作用下Birkhoff系统Lie对称性的摄动，得到了系统的一类新型绝热不变量.举例说明结果的应用. 关键词： Birkhoff系统 Lie对称性 摄动 绝热不变量     

相对论性Birkhoff系统的对称性摄动及其逆问题

研究在小干扰力作用下相对论性Birkhoff系统的对称性摄动问题.建立了相对论性Birkhoff系统的基本原理、运动方程和小扰动方程.讨论该系统的Lie对称性变换和守恒量.研究在无限小变换下该系统的对称性摄动，构造了s阶绝热不变量.给出了绝热不变量存在的条件和形式.研究该系统的对称性摄动逆问题，当系统存在s阶绝热不变量时，得到了该系统的无限小变换的对称性摄动.研究相对论性Birkhoff系统和经典Birkhoff系统对称性摄动之间的关系. 关键词： Lie对称性 摄动 绝热不变量 相对论