经典力学中加速度相关的Lagrange函数

研究了加速度线性相关的Lagrange函数,在加速度项系数对称的条件下,Lagrange方程保持为二阶微分方程;给出了从运动方程构造加速度相关的Lagrange函数的方法;研究同一系统的加速度相关和加速度无关的Lagrange函数之间的关系.举例说明结果的应用. 关键词： Lagrange方程 加速度相关的Lagrange函数 广义力学 Lagrange函数的规范变换     

一维变系数耗散系统Lagrange函数和Hamilton函数的新构造方法

提出构造二阶微分方程的Lagrange函数和Hamilton函数的新路径. 将二阶方程写成一阶方程组并构造出对应的一阶Lagrange函数后,直接从一阶Lagrange函数导出二阶Lagrange函数和Hamilton函数. 利用上述方法得到若干耗散和类耗散系统的一阶和二阶Lagrange函数以及Hamilton函数;讨论了这种方法的优点. 举例说明所得结果的应用. 关键词： 逆问题 耗散系统 Lagrange函数 Hamilton函数     

关于Noether对称性的两种理解

介绍了对Lagrange系统Noether对称性的两种理解，一种理解为Lagrange函数的不变性，另一种理解为作用量的不变性.研究表明，这两种理解是不同的.在一些条件下，Lagrange函数的不变性可以成为作用量的不变性，在另一些条件下，作用量的不变性可以成为Lagrange函数的不变性.将Noether对称性理解为作用量的不变性是合理的. 关键词： Lagrange系统 Noether对称性 作用量的不变性 Lagrange函数的不变性     

完整有势力学系统的高阶Lagrange方程

首先提出力学系统高阶速度能定理,阐明了系统高阶速度能量的物理意义;然后提出力学系统有势的一般判据.在此基础上,引入高阶Lagrange函数,得出完整有势力学系统的高阶Lagrange方程,并得到系统高阶循环积分和高阶广义能量积分. 关键词： 高阶速度能定理 有势力学系统 高阶Lagrange方程 高阶Lagrange函数     

计算加速度相关Lagrange函数的方法

研究从运动微分方程构造加速度相关Lagrange函数的问题,给出一种由方程的自伴随形式计算加速度相关Lagrange函数的方法.举例说明结果的应用. 关键词： 分析力学 加速度相关Lagrange函数 自伴随性 Lagrange力学逆问题     

一阶Lagrange系统的梯度表示

研究一阶Lagrange系统的梯度表示. 给出一阶Lagrange系统可成为梯度系统的条件. 利用梯度系统的性质研究系统的稳定性. 给出例子说明结果的应用. 关键词： 一阶Lagrange系统 梯度系统 稳定性     

相对运动动力学系统的Lagrange对称性

研究相对运动动力学系统的Lagrange对称性.给出系统Lagrange对称性的定义和判据.找到这类对称性导致的守恒量的条件和形式.举例说明结果的应用. 关键词： 相对运动动力学 Lagrange对称性 守恒量     

Lagrange系统在施加陀螺力后的对称性

研究Lagrange系统在施加陀螺力后的Noether对称性与Lie对称性.给出系统在施加陀螺力后 ，可保持其Noether对称性与Noether守恒量的条件.给出系统在施加陀螺力后，可保持其Lie 对称性与Hojman守恒量的条件.最后，举例说明结果的应用. 关键词： Lagrange系统 陀螺力 对称性 守恒量     

非Chetaev型非完整系统的Lagrange对称性与守恒量

研究非Chetaev型非完整非保守力学系统的Lagrange对称性.给出了系统的Lagrange对称性的定义和判据,得到了非Chetaev型非完整非保守力学系统的Lagrange对称性导致守恒量（第一积分)的条件及其形式.举例说明结果的应用. 关键词： 非Chetaev型非完整约束 Lagrange对称性 守恒量     

变质量非完整系统的Lagrange对称性与守恒量

本文研究变质量非完整系统的Lagrange对称性, 给出变质量非完整系统Lagrange对称性的判据, 得到变质量非完整系统Lagrange对称性导致的守恒量及其存在的条件, 并举例说明结果的应用.     

LEPAGE EQUIVALENTS OF SECOND-ORDER EULER–LAGRANGE FORMS AND THE INVERSE PROBLEM OF THE CALCULUS OF VARIATIONS

In the calculus of variations, Lepage (n + 1)-forms are closed differential forms, representing Euler–Lagrange equations. They are fundamental for investigation of variational equations by means of exterior differential systems methods, with important applications in Hamilton and Hamilton–Jacobi theory and theory of integration of variational equations. In this paper, Lepage equivalents of second-order Euler–Lagrange quasi-linear PDE's are characterised explicitly. A closed (n + 1)-form uniquely determined by the Euler–Lagrange form is constructed, and used to find a geometric solution of the inverse problem of the calculus of variations.     

基于Lagrange方法封闭的两相湍流场方程模型

两相湍流场方程模型采用基于Euler方法的一阶矩方程,而二阶矩方程由Lagrange方法得到,新模型的封闭不需要附加其它假设.首先基于概率密度函数给出颗粒运动的连续方程和动量方程,其次由基于平均Langevin方程的Lagrange模型推导得到颗粒二阶矩方程,最终获得封闭的二阶矩模型.将新模型用于气固两相壁面射流的数值模拟,结果表明新模型合理有效.     

Lagrange--Noether method for solving second-order differential equations

The purpose of this paper is to provide a new method called the Lagrange--Noether method for solving second-order differential equations. The method is, firstly, to write the second-order differential equations completely or partially in the form of Lagrange equations, and secondly, to obtain the integrals of the equations by using the Noether theory of the Lagrange system. An example is given to illustrate the application of the result.     

Discrete Lagrangian Reduction,Discrete Euler–Poincaré Equations,and Semidirect Products

A discrete version of Lagrangian reduction is developed within the context of discrete time Lagrangian systems on G × G, where G is a Lie group. We consider the case when the Lagrange function is invariant with respect to the action of an isotropy subgroup of a fixed element in the representation space of G. Within this context, the reduction of the discrete Euler–Lagrange equations is shown to lead to the so-called discrete Euler–Poincaré equations. A constrained variational principle is derived. The Legendre transformation of the discrete Euler–Poincaré equations leads to discrete Hamiltonian (Lie–Poisson) systems on a dual space to a semiproduct Lie algebra.     

An inverse problem in analytical dynamics

This paper presents an inverse problem in analytical dynamics. The inverse problem is to construct the Lagrangian when the integrals of a system are given. Firstly, the differential equations are obtained by using the time derivative of the integrals. Secondly, the differential equations can be written in the Lagrange equations under certain conditions and the Lagrangian can be obtained. Finally, two examples are given to illustrate the application of the result.     

完整力学系统的高阶运动微分方程

从质点系的牛顿动力学方程出发,引入系统的高阶速度能量,导出完整力学系统的高阶Lagrange方程、高阶Nielsen方程以及高阶Appell方程,并证明了完整系统三种形式的高阶运动微分方程是等价的.结果表明,完整系统高阶运动微分方程揭示了系统运动状态的改变与力的各阶变化率之间的联系,这是牛顿动力学方程以及传统分析力学方程不能直接反映的.因此,完整系统高阶运动微分方程是对牛顿动力学方程及传统Lagrange方程、Nielsen方程、Appell方程等二阶运动微分方程的进一步补充. 关键词： 高阶速度能量 高阶Lagrange方程 高阶 Nielsen方程 高阶Appell方程     

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.     

Second-order integrable Lagrangians and WDVV equations

Letters in Mathematical Physics - We investigate the integrability of Euler–Lagrange equations associated with 2D second-order Lagrangians of the form $\begin{aligned} \int...     

Chebyshev pseudospectral solution of second-order elliptic equations with finite element preconditioning

A new Chebyshev pseudospectral algorithm for second-order elliptic equations using finite element preconditioning is proposed and tested on various problems. Bilinear and biquadratic Lagrange elements are considered as well as bicubic Hermite elements. The numerical results show that bilinear elements produce spectral accuracy with the minimum computational work. L-shaped regions are treated by a subdomain approach.