Lie symmetry algebra of one-dimensional nonconservative dynamical systems

Lie symmetry algebra of linear nonconservative dynamical systems is studied in this paper. By using 1--1 mapping, the Lie point and Lie contact symmetry algebras are obtained from two independent solutions of the one-dimensional linear equations of motion.     

李代数方法在分子振动光谱研究中的应用

通过对动力学对称性的介绍,引出构造分子体系Hamiltonian的方法,并介绍李代数方法处理分子"振动-振动耦合"、"振动-转动耦合"Hamiltonian的一般步骤.还以HCO分子体系为例,说明如何选取特定的群链来构造Hamiltonian,并论证了对于该体系群链的唯一性.最后还对一些李代数方法在分子光谱中的其他应用(例如局域模,多原子分子的振转光谱研究)作了初步的介绍,并对于这一方法的不足和局限性做了论述.     

Lie symmetries and non-Noether conserved quantities for Hamiltonian canonical equations

This paper focuses on studying Lie symmetries and non-Noether conserved quantities of Hamiltonian dynamical systems in phase space. Based on the infinitesimal transformations with respect to the generalized coordinates and generalized momenta, we obtain the determining equations and structure equation of the Lie symmetry for Hamiltonian dynamical systems. This work extends the research of non-Noether conserved quantity for Hamilton canonical equations, and leads directly to a new type of non-Noether conserved quantities of the systems. Finally, an example is given to illustrate these results.     

Lie symmetries and invariants of constrained Hamiltonian systems

According to the theory of the invariance of ordinary differential equations under the infinitesimal transformations of group, the relations between Lie symmetries and invariants of the mechanical system with a singular Lagrangian are investigated in phase space. New dynamical equations of the system are given in canonical form and the determining equations of Lie symmetry transformations are derived. The proposition about the Lie symmetries and invariants are presented. An example is given to illustrate the application of the result in this paper.     

离散Lagrange系统的Lie对称性

研究离散Lagrange系统的Lie对称性. 根据离散变分原理建立离散系统的运动方程. 给出离散运动方程Lie对称性的定义和确定方程. 举例说明结果的应用. 关键词： 离散Lagrange系统 离散变分原理 Lie对称性 确定方程     

量子变换的李代数方法

利用李代数生成元构成幺正算符,对耦合玻色体系给出了一种简单的对角化方法.     

Lie symmetries and exact solutions for a short-wave model

In this paper, the Lie symmetry analysis and generalized symmetry method are performed for a short-wave model. The symmetries for this equation are given, and the phase portraits of the traveling wave systems are analyzed using the bifurcation theory of dynamical systems. The exact parametric representations of four types of traveling wave solutions are obtained.     

推广的一类Lie代数及其相关的一族可积系统

对已知的Lie代数An-1作直接推广得到一类新的Lie代数gl(n,C).为应用方便，本文只考虑Lie代数gl(3,C)情形.构造了gl(3,C)的一个子代数，通过对阶数的规定，得到了一类新的loop代数.作为其应用，设计了一个新的等谱问题，得到了一个新的Lax对.利用屠格式获得了一族新的可积系统，具有双Hamilton结构,且是Liouville可积系.作为该方程族的约化情形，得到了新的耦合广义Schrdinger方程. 关键词： Lie代数 可积系 Hamilton结构     

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     

Lie symmetries and conserved quantities of controllable nonholonomic dynamical systems

This paper concentrates on studying the Lie symmetries and conserved quantities of controllable nonholonomicdynamicM systems. Based on the infinitesimal transformation, we establish the Lie symmetric determining equationsand restrictive equations and give three definitions of Lie symmetries before the structure equations and conservedquantities of tile Lie symmetries are obtained. Then we make a study of the inverse problems. Finally, an example ispresented for illustrating the results.     

Mei Symmetry and Lie Symmetry of Relativistic Hamiltonian System

The Mei symmetry and the Lie symmetry of the relativistic Hamiltonian system are studied. The definition and criterion of the Mei symmetry and the Lie symmetry of the relativistic Hamiltonian system are given. The relationship between them is found. The conserved quantities which the Mei symmetry and the Lie symmetry lead to are obtained.An example is given to illustrate the application of the result.     

Differential geometry of the Lie algebra of the quantum plane

We present a differential calculus on the extension of the quantum plane obtained by considering that the (bosonic) generator x is invertible and by working with polynomials in ln x instead of polynomials in x. We construct the quantum Lie algebra associated with this extension and obtain its Hopf algebra structure and its dual Hopf algebra.     

相对论力学系统的形式不变性与Lie对称性

研究相对论力学系统的形式不变性和Lie对称性.给出相对论力学系统在无限小变换下形式不变性和Lie对称性的定义、判据和守恒量,得到形式不变性和Lie对称性的关系,并举例说明结果的应用. 关键词： 相对论 力学系统 形式不变性 Lie对称性     

Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems

This paper focuses on studying Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems. Firstly, the discrete generalized Hamiltonian canonical equations and discrete energy equation of nonholonomic Hamiltonian systems are derived from discrete Hamiltonian action. Secondly, the determining equations and structure equation of Lie symmetry of the system are obtained. Thirdly, the Lie theorems and the conservation quantities are given for the discrete nonholonomic Hamiltonian systems. Finally, an example is discussed to illustrate the application of the results.     

Lie Point Symmetries of Differential-Difference Equations

In this paper, the classical Lie group approach is extended to find some Lie point symmetries of differentialdifference equations. It reveals that the obtained Lie point symmetries can constitute a Kac-Moody-Virasoro algebra.     

Mei Symmetry and Lie Symmetry of Relativistic Hamiltonian System

The Mei symmetry and the Lie symmetry of the relativistic Hamiltonian system are studied. The definition and criterion of the Mei symmetry and the Lie symmetry of the reoativistic Hamiltonian system are given. The relationship between them is found. The conserved quantities which the Mei symmetry and the Lie symmetry lead to are obtained. An example is given to illustrate the application of the result.     

一类非完整奇异系统的Lie对称性与守恒量

利用微分方程在无限小变换群下的不变性,研究一类非完整奇异系统的Lie对称性.给出Lie对称性的确定方程、限制方程、附加限制方程和结构方程,并给出守恒量的形式 关键词： 奇异系统 非完整约束 Lie对称性 守恒量     

有限自由度系统的定域Lie对称性和守恒量

研究有限自由度系统定域Lie对称性的正问题和逆问题。给出了定域Lie对称性的定义、确定方程、结构方程和守恒量。进一步从定域Lie对称性导出在有限连续群无限小变换下的Lie对称性。最后给出一个例子去说明这些结果。     

Study of the Lie symmetries of a relativistic variable mass system

The differential equations of motion of a relativistic variable mass system are given.By using the invariance of the differential equations under the infinitesimal transformations of groups,the determining equations and the restriction equations of the Lie symmetries of a relativistic variable mass system are built,and the structure equation and the conserved quantity of the Lie symmetries are obtained.Then the inverse problem of the Lie symmetries is studied.The corresponding Lie symmetries are found according to a known conserved quantity.An example is given to illustrate the application of the result.