一类动力学方程的Mei对称性

研究一类动力学方程的Mei对称性的定义和判据，由Mei对称性通过Noether对称性可找到Noether守恒量.由Mei对称性通过Lie对称性可找到Hojman守恒量.同时，也可找到一类新型守恒量.     

Emden方程的Mei对称性、Lie对称性和Noether对称性

研究Emden动力学方程的形式不变性即Mei对称性，给出其定义和确定方程，研究Emden方程的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，给出一个例子说明结果的应用. 关键词： Emden动力学方程 Mei对称性 Noether对称性 Lie对称性     

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     

Lie symmetry and Mei conservation law of continuum system

Lie symmetry and Mei conservation law of continuum Lagrange system are studied in this paper.The equation of motion of continuum system is established by using variational principle of continuous coordinates.The invariance of the equation of motion under an infinitesimal transformation group is determined to be Lie-symmetric.The condition of obtaining Mei conservation theorem from Lie symmetry is also presented.An example is discussed for applications of the results.     

Theory of symmetry for a rotational relativistic Birkhoff system

The theory of symmetry for a rotational relativistic Birkhoff system is studied. In terms of the invariance of the rotational relativistic Pfaff－Birkhoff－D'Alembert principle under infinitesimal transformations, the Noether symmetries and conserved quantities of a rotational relativistic Birkhoff system are given. In terms of the invariance of rotational relativistic Birkhoff equations under infinitesimal transformations, the Lie symmetries and conserved quantities of the rotational relativistic Birkhoff system are given.     

Kepler方程的共形不变性、Mei对称性与守恒量

研究Kepler系统在无限小变换下的共形不变性、Mei对称性.给出该系统与总能量、角动量不同的新守恒量.并在广义坐标和广义速度构成的空间中讨论这些守恒量的独立性.     

General criterion of invariance forintegro-differential equations

General criterion of invariance of integro-differential equations under Lie symmetry groupof point transformations is derived. It is a generalization of the previous form of the criterion to the case of a moving range of integration. This is the situation when a region of integration depends on external, with respect to integration, variables which leads to its explicit dependence on a group parameter.     

The construction of conserved quantities for linearly coupled oscillators and study of symmetries about the conserved quantities

In this paper, the conserved quantities are constructed using two methods. The first method is by making an ansatz of the conserved quantity and then using the definition of Poisson bracket to obtain the coefficients in the ansatz. The main procedure for the second method is given as follows. Firstly, the coupled terms in Lagrangian are eliminated by changing the coordinate scales and rotating the coordinate axes, secondly, the conserved quantities are obtain in new coordinate directly, and at last, the conserved quantities are expressed in the original coordinates by using the inverse transform of the coordinates. The Noether symmetry and Lie symmetry of the infinitesimal transformations about the conserved quantities are also studied in this paper.     

Lie symmetry and Mei continuum conservation law of system＊

Lie symmetry and Mei conservation law of continuum Lagrange system are studied in this paper. The equation of motion of continuum system is established by using variational principle of continuous coordinates. The invariance of the equation of motion under an infinitesimal transformation group is determined to be Lie-symmetric. The condition of obtaining Mei conservation theorem from Lie symmetry is also presented. An example is discussed for applications of the results.     

AN OLD METHOD OF JACOBI TO FIND LAGRANGIANS

In a recent paper by Ibragimov a method was presented in order to find Lagrangians of certain second-order ordinary differential equations admitting a two-dimensional Lie symmetry algebra. We present a method devised by Jacobi which enables one to derive (many) Lagrangians of any second-order differential equation. The method is based on the search of the Jacobi Last Multipliers for the equations. We exemplify the simplicity and elegance of Jacobi's method by applying it to the same two equations as Ibragimov did. We show that the Lagrangians obtained by Ibragimov are particular cases of some of the many Lagrangians that can be obtained by Jacobi's method.     

Classical and nonclassical symmetries analysis for initial value problems

Classical and nonclassical symmetries are considered to reduce evolution equations with initial conditions in two independent variables. First of all, we rearrange the classical infinitesimal operators such that they leave the initial value problems invariant. Secondly, we give a sufficient condition for the nonclassical symmetry reductions of initial value problems. The generalized Kuramoto-Sivashinsky equation with dispersive effects is considered to examine the algorithms.     

Lie point symmetry algebras and finite transformation groups of the general Broer--Kaup system

Using a new symmetry group theory, the transformation groups and symmetries of the general Broer--Kaup system are obtained. The results are much simpler than those obtained via the standard approaches.     

Noether symmetry and Lie symmetry of discrete holonomic systems with dependent coordinates

The Noether symmetry, the Lie symmetry and the conserved quantity of discrete holonomic systems with dependent coordinates are investigated in this paper. The Noether symmetry provides a discrete Noether identity and a conserved quantity of the system. The invariance of discrete motion equations under infinitesimal transformation groups is defined as the Lie symmetry, and the condition of obtaining the Noether conserved quantity from the Lie symmetry is also presented. An example is discussed to show the applications of the results.     

1-Soliton Solution of the Nonlinear Schrödinger’s Equation with Kerr Law Nonlinearity Using Lie Symmetry Analysis

This paper obtains the 1-soliton solution of the nonlinear Schrödinger’s equation in a Kerr law media. The technique that is used to carry out the integration of this equation is the Lie symmetry analysis.     

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.     

广义经典力学系统的对称性与Mei守恒量

研究广义经典力学系统的对称性和一类新型守恒量——Mei守恒量.在高维增广相空间中建立 了系统的运动微分方程；给出了系统的Mei对称性、Noether对称性和Lie对称性的判据；得 到了分别由三种对称性导致Mei守恒量的条件和Mei守恒量的形式.举例说明结果的应用. 关键词： 广义经典力学 Mei对称性 Noether对称性 Lie对称性 守恒量     

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.     

Classification and Approximate Solutions to a Class of Perturbed Nonlinear Wave Equations

A complete approximate symmetry classification of a class of perturbed nonlinear wave equations is performed using the method originated from Fushchich and Shtelen. Moreover, large classes of approximate invariant solutions of the equations based on the Lie group method are constructed.     

Symmetries and Mei Conserved Quantities of Nonholonomic Controllable Mechanical Systems

This paper concentrates on studying the symmetries and a new type of conserved quantities called Mei conserved quantity. The criterions of the Mei symmetry, the Noether symmetry and the Lie symmetry are given. The conditions and the forms of the Mei conserved quantities deduced from these three symmetries are obtained. An example is given to illustrate the application of the result.