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     

完整系统三阶Lagrange方程的一种推导与讨论

从牛顿运动方程出发，推导了完整系统关于广义加速度的Lagrange方程.讨论了该方程与传统分析力学中的Lagrange方程的相容性问题.结果显示，三阶Lagrange方程可以通过对Lagrange方程求一阶时间导数得到，表明它们是相容的.因此三阶Lagrange方程提供了一种不同于传统Lagrange方程方法的求解物体运动方程的途径. 关键词： Lagrange方程 加速度能量 广义坐标     

关于Birkhoff表示的Lagrange像的研究

研究运动微分方程Birkhoff表示的Lagrange像.得出二阶Lagrange函数应满足的条件,在此条件下广义Lagrange方程为二阶微分方程组;提出新的求解Lagrange力学逆问题路线;指出在此问题研究中曾发生过的失误.举例说明所得结果的应用.     

阻尼落体运动的分析力学研究

应用分析力学理论和方法,研究了两种情况下的阻尼落体运动:1)阻力大小与速度成正比;2)阻力大小与速度平方成正比.对两种运动分别给出了等效的Lagrange函数和Hamilton函数,并应用第一积分法、点变换法、正则变换法和Ham-ilton-Jacobi方程法等不同的求解方法进行了求解.     

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     

线性三原子分子振动的Lagrange函数和Hamilton函数

本文以线性耦合振子为模型,导出了三种不同坐标变量表示下线性对称三原子分子的振动微分方程,利用分析力学逆问题理论和方法,构造出了五种对应的Lagrange函数和Hamilton函数,其中有些是新的结果.     

阻尼耦合振动的Lagrange函数和Hamilton函数

本文研究两种阻尼耦合振动的分析力学化.首先,利用坐标变换将方程变换成自伴随形式;其次,根据Engels方法计算得到Lagrange函数;最后,由逆变换导出原始方程的Lagrange函数,以及Hamilton函数.     

磁场中带电粒子阻尼运动的分析力学表示

研究带电粒子在磁场中作阻尼运动的分析力学表示. 首先, 求解运动微分方程的Birkhoff力学逆问题, 得到带电粒子的4个Rirkhoff表示; 其次, 导出4个状态空间中Lagrange表示和对应的4个位形空间中Lagrange表示; 第三, 构造出4个Hamilton函数; 最后, 从粒子运动的分析力学表示直接得到4个第一积分, 并求出运动方程的解.     

一类Painleve方程的Lagrange函数族

利用第一积分构造Lagrange函数的理论和方法, 导出一类Painleve方程的两个Lagrange函数族, 以及一些Lagrange函数和Hamilton函数.     

A Note on the Equivalence of Post-Newtonian Lagrangian and Hamiltonian Formulations

Recently,it has been generally claimed that a low order post-Newtonian(PN)Lagrangian formulation,whose Euler-Lagrange equations are up to an infinite PN order,can be identical to a PN Hamiltonian formulation at the infinite order from a theoretical point of view.In general,this result is difficult to check because the detailed expressions of the Euler-Lagrange equations and the equivalent Hamiltonian at the infinite order are clearly unknown.However,there is no difficulty in some cases.In fact,this claim is shown analytically by means of a special first-order post-Newtonian(1PN)Lagrangian formulation of relativistic circular restricted three-body problem,where both the Euler-Lagrange equations and the equivalent Hamiltonian are not only expanded to all PN orders,but have converged functions.It is also shown numerically that both the Euler-Lagrange equations of the low order Lagrangian and the Hamiltonian are equivalent only at high enough finite orders.     

Canonical symmetry properties of the constrained singular generalized mechanical system

Based on generalized Apell－Chetaev constraint conditions and to take the inherent constrains for singular Lagrangian into account, the generalized canonical equations for a general mechanical system with a singular higher-order Lagrangian and subsidiary constrains are formulated. The canonical symmetries in phase space for such a system are studied and Noether theorem and its inversion theorem in the generalized canonical formalism have been established.     

Determination of the Electromagnetic Lagrangian from a System of Poisson Brackets

The Lagrangian and Hamiltonian formulations of electromagnetism are reviewed and the Maxwell equations are obtained from the Hamiltonian for a system of many electric charges. It is shown that three of the equations which were obtained from the Hamiltonian, namely the Lorentz force law and two Maxwell equations, can be obtained as well from a set of postulated Poisson brackets. It is shown how the results derived from these brackets can be used to reconstruct the original Lagrangian for the theory aided by some reasoning based on physical concepts.     

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.     

Higher-Order Lagrangian Equations of Higher-Order Motive Mechanical System

In this paper, if the condition of variation δt=0 is satisfied, the higher-order Lagrangian equations and higher-order Hamilton's equations, which show the consistency with the results of traditional analytical mechanics, are obtained from the higher-order Lagrangian equations and higher-order Hamilton's equations. The results can enrich the theory of analytical mechanics.     

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.     

The shallow water equations in Lagrangian coordinates

Recent advances in the collection of Lagrangian data from the ocean and results about the well-posedness of the primitive equations have led to a renewed interest in solving flow equations in Lagrangian coordinates. We do not take the view that solving in Lagrangian coordinates equates to solving on a moving grid that can become twisted or distorted. Rather, the grid in Lagrangian coordinates represents the initial position of particles, and it does not change with time. We apply numerical methods traditionally used to solve differential equations in Eulerian coordinates, to solve the shallow water equations in Lagrangian coordinates. The difficulty with solving in Lagrangian coordinates is that the transformation from Eulerian coordinates results in solving a highly nonlinear partial differential equation. The non-linearity is mainly due to the Jacobian of the coordinate transformation, which is a precise record of how the particles are rotated and stretched. The inverse Jacobian must be calculated, thus Lagrangian coordinates cannot be used in instances where the Jacobian vanishes. For linear (spatial) flows we give an explicit formula for the Jacobian and describe the two situations where the Lagrangian shallow water equations cannot be used because either the Jacobian vanishes or the shallow water assumption is violated. We also prove that linear (in space) steady state solutions of the Lagrangian shallow water equations have Jacobian equal to one. In the situations where the shallow water equations can be solved in Lagrangian coordinates, accurate numerical solutions are found with finite differences, the Chebyshev pseudospectral method, and the fourth order Runge–Kutta method. The numerical results shown here emphasize the need for high order temporal approximations for long time integrations.     

Noether Symmetry of Three-Order Lagrangian Equations

Based on the three-order Lagrangian equation, pseudo-Hamilton actoon I^＊ is defined and the three-order Hamilton＇s principle and the conditions are obtained in the paper. Then, the Noether symmetry about three-order Lagrangian equations is deduced. Finally, an example is given to illustrate the application of the result.