Conformal invariance and Hojman conserved quantities of first order Lagrange systems

In this paper the conformal invariance by infinitesimal transformations of first order Lagrange systems is discussed in detail. The necessary and sufficient conditions of conformal invariance and Lie symmetry simultaneously by the action of infinitesimal transformations are given. Then it gets the Hojman conserved quantities of conformal invariance by the infinitesimal transformations. Finally 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.

A new type of conserved quantity of Lie symmetry for the Lagrange system

This paper studies a new type of conserved quantity which is directly induced by Lie symmetry of the Lagrange system. Firstly, the criterion of Lie symmetry for the Lagrange system is given. Secondly, the conditions of existence of the new conserved quantity as well as its forms are proposed. Lastly, an example is given to illustrate the application of the result.

A new type of conserved quantity of Mei symmetry for Lagrange system

This paper studies a new type of conserved quantity which is directly induced by Mei symmetry of the Lagrange system. Firstly, the definition and criterion of Mei symmetry for the Lagrange system are given. Secondly, a coordination function is introduced, and the conditions of existence of the new conserved quantity as well as its forms are proposed. Lastly, an illustrated example is given. The result indicates that the coordination function can be selected properly according to the demand for finding the gauge function, and thereby the gauge function can be found more easily. Furthermore, since the choice of the coordination function has multiformity, many more conserved quantities of Mei symmetry for the Lagrange system can be obtained.

Conformal invariance and conserved quantities of non-conservative Lagrange systems by point transformations

This paper studies the conformal invariance by infinitesimal point transformations of non-conservative Lagrange systems. It gives the necessary and sufficient conditions of conformal invariance by the action of infinitesimal point transformations being Lie symmetric simultaneously. Then the Noether conserved quantities of conformal invariance are obtained. Finally an illustrative example is given to verify the results.

Hojman conserved quantity deduced by weak Noether symmetry for Lagrange systems

This paper studies the Hojman conserved quantity, a non-Noether conserved quantity, deduced by special weak Noether symmetry for Lagrange systems. Under special infinitesimal transformations in which the time is not variable, its criterion is given and a method of how to seek the Hojman conserved quantity is presented. A Hojman conserved quantity can be found by using the special weak Noether symmetry.

Noether's theory of Lagrange systems in discrete case

In this paper, Noether theory of Lagrange systems in discrete case are studied. First, we briefly overview the wellknown Noether theory of Lagrange system in the continuous case. Then, we introduce some definitions and notations, such as the operators of discrete translation to the right and the left and the operators of discrete differentiation to the right and the left, and give the conditions for the invariance of the difference functional on the uniform lattice and the non-uniform one, respectively. We also deduce the discrete analog of the Noether-type identity. Finally, the discrete analog of Noether＇s theorem is presented. An example was discussed to illustrate these results.

Lie symmetries, Perturbation to symmetries and adiabatic invariants of Lagrange system

作者为我校物理与信息工程系陈向炜教授．全文发表于SCI源期刊《PhysicsLettersA》2005年第337期．该文属陈向炜教授主持的国家自然科学基金（批准号10372053）和河南省自然科学基金（批准号0311010900）资助课题．

Dynamic event-triggered finite-time control for multiple Euler-Lagrange systems using integral terminal sliding mode

The tracking control for multiple Euler-Lagrange systems with external disturbances in finite time under undirected topology is investigated in this paper.A dynamic model is established for the multi-EL systems to accurately describe the general mechanical system.Furthermore,an integral terminal sliding mode surface is devised to converge the tracking errors of the system state to a neighborhood of zero within finite time,and the designed finite-time controller ensures fast convergence and high steady-state accuracy.To reduce the controller update frequency and network transmission communication load,a dynamic event-triggered mechanism is introduced between the sensor and controller,and no Zeno behavior was observed.Therefore,the Lyapunov stability theory and finite-time stability criterion prove that all signals in the closed-loop system are uniformly ultimately bounded in finite time.Finally,the simulation results verified the effectiveness of the proposed control method.

基于LAGRANGE方程的深水钻井隔水管–水下井口系统动力分析

为了探究大质量、大刚度防喷器组(blowout preventers,BOPs)对深水钻井隔水管系统动态相应预测精度的影响,根据细长钻井隔水管与刚性防喷器组的结构特点,提出两者刚柔耦合的概念,采用能量法推导隔水管–防喷器组–水下井口系统的动能和势能,采用LAGRANGE方法建立耦合系统动力学理论模型,采用科学计算软件和Newmark-β直接积分法对动力学模型进行数值计算。以南海某深水钻井隔水管为例,开展基于耦合动力学模型的隔水管系统动态响应分析。结果表明,采用本文理论模型得到的隔水管不同位置的节点位移、单元弯矩、上部和下部挠性接头转角时程曲线、整体侧向位移包络线和弯矩包络线等与ABAQUS仿真结果均吻合良好,最大误差为8.8%。此方法可为深水钻井隔水管和水下井口系统动态分析提供参考。

常微分方程组初值问题的Lagrange插值逼近方法

常微分方程的快速发展和很多学科有着紧密的关系,随着众多的数学家对其研究的不断加深,极大地推动了现代数学的发展。因此,本文研究拉格朗日插值(Lagrange)逼近方法在常微分方程组初值问题上的应用,推导出Lagrange插值法的逼近算法,求解常微分方程组初值问题的数值解,也就是通过具体的例子,利用Lagrange插值逼近方法构造相对应的数值格式,寻找误差和多项式次数的关系,将结果可视化,最后对相应的误差结果进行分析。数值结果表明Lagrange插值逼近方法具有较高的精度。

A Modified Lagrange Method for Solving Convex Quadratic Optimization Problems

In this paper, a modified version of the Classical Lagrange Multiplier method is developed for convex quadratic optimization problems. The method, which is evolved from the first order derivative test for optimality of the Lagrangian function with respect to the primary variables of the problem, decomposes the solution process into two independent ones, in which the primary variables are solved for independently, and then the secondary variables, which are the Lagrange multipliers, are solved for, afterward. This is an innovation that leads to solving independently two simpler systems of equations involving the primary variables only, on one hand, and the secondary ones on the other. Solutions obtained for small sized problems (as preliminary test of the method) demonstrate that the new method is generally effective in producing the required solutions.

爆轰驱动多介质问题的Lagrange多分区自适应数值模拟研究

凝聚炸药爆轰驱动惰性金属材料形成的多介质流动问题广泛存在于工程应用领域,Lagrange方法由于其物质界面的高保真特性一直在相关问题的数值模拟中发挥着不可替代的作用.加密网格是提高爆轰驱动多介质问题模拟精度的常用途径之一.然而,Lagrange框架下整体密网格计算常会遇到网格畸变、计算效率低等问题.为此,针对爆轰驱动多介质流动问题,提出了一种Lagrange框架下的非结构网格多层自适应方法,在保证所关心区域局部网格分辨率的前提下,大幅缩减了整体计算规模,提升了Lagrange计算的健壮性.设计了非结构网格多层数据结构,提出了多层网格分层存储、有效网格压至一层进行Lagrange计算的AMR策略,同时还发展了自适应接触滑移耦合算法,实现了AMR计算与多分区接触滑移计算的“紧耦合”.相比于已有工作,所提出的AMR方法既保持了非结构网格多层数据结构的灵活性优势,又避免了Lagrange框架下多层网格分别计算带来的层间耦合困难,同时因实现了与接触滑移的自适应耦合,使得它能很好地适应多分区的多介质问题.在一维、二维爆轰算例验证所提出方法正确性的前提下,开展了拐角爆轰、多层炸药隔爆和有限尺寸弯道爆轰等复杂爆轰弹塑性多介质问题的数值模拟研究.计算结果显示,采用Lagrange非结构多层AMR数值模拟方法,可以有效捕捉凝聚炸药爆轰与惰性金属材料相互作用过程中的波系结构,可节省90%以上的网格规模.该方法对复杂几何边界、复杂波系结构和多分区相互作用等问题具有良好的适应性,为后续开展凝聚炸药爆轰约束问题的机理研究奠定了坚实的基础.

Lagrange equations of nonholonomic systems with fractional derivatives

This paper obtains Lagrange equations of nonholonomic systems with fractional derivatives. First, the exchanging relationships between the isochronous variation and the fractional derivatives are derived. Secondly, based on these exchanging relationships, the Hamilton＇s principle is presented for non-conservative systems with fractional derivatives. Thirdly, Lagrange equations of the systems are obtained. Furthermore, the d＇Alembert-Lagrange principle with fractional derivatives is presented, and the Lagrange equations of nonholonomic systems with fractional derivatives are studied. An example is designed to illustrate these results.

Discrete Fractional Lagrange Equations of Nonconservative Systems

In order to study discrete nonconservative system,Hamilton's principle within fractional difference operators of Riemann-Liouville type is given. Discrete Lagrange equations of the nonconservative system as well as the nonconservative system with dynamic constraint are established within fractional difference operators of Riemann-Liouville type from the view of time scales. Firstly,time scale calculus and fractional calculus are reviewed.Secondly,with the help of the properties of time scale calculus,discrete Lagrange equation of the nonconservative system within fractional difference operators of Riemann-Liouville type is presented. Thirdly,using the Lagrange multipliers,discrete Lagrange equation of the nonconservative system with dynamic constraint is also established.Then two special cases are discussed. Finally,two examples are devoted to illustrate the results.

Conformal invariance and conserved quantity of third-order Lagrange equations for non-conserved mechanical systems

This paper studies conformal invariance and conserved quantity of third-order Lagrange equations for non- conserved mechanical systems. Third-order Lagrange equations, the definition and a determining equation of conformal invariance of the system are presented. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The necessary and sufficient condition that conformal invaxiance of the system would have Lie symmetry under single-parameter infinitesimal transformations is obtained. The corresponding conserved quantity of conformal invariance is derived with the aid of a structure equation. Lastly, an example is given to illustrate the application of the results.

时间分数阶Black-Scholes方程的重心Lagrange插值配点法

针对欧式期权定价的时间分数阶Black-Scholes模型,设计一种重心Lagrange插值配点法格式.首先,采用Laplace变换近似Caputo型分数阶导数,将分数阶方程转化为整数阶方程;然后,在时-空方向上均采用重心Lagrange插值配点法进行离散,构造重心Lagrange插值配点法格式.结果表明:时间分数阶Black-Scholes方程的重心Lagrange插值配点法具有高精度和有效性.

基于非标准Lagrange函数的分数阶Lagrange系统的Noether对称性与守恒量

研究基于两类非标准Lagrange函数(指数Lagrange函数和幂律Lagrange函数)的分数阶Lagrange系统的Noether对称性与守恒量.首先,分别导出Caputo分数阶导数下两类非标准Lagrange系统的运动微分方程;其次,根据作用量在无穷小变换下的不变性,给出了分数阶非标准Lagrange系统的Noether对称变换的定义和判据;最后,建立系统的Noether定理并举例说明结果的应用.

Capacity Optimization Using Augmented Lagrange Method in Intelligent Reflecting Surface-Based MIMO Communication Systems

In this paper,we present an algorithm for capacity optimization in intelligent reflecting surface(IRS)-based multiple-input multiple-output(MIMO)communication systems.To maximize the capacity of elements in IRS,we use augmented Lagrange method with the equivalent transformations on the covariance matrix and reflection matrix constraints.This results an adjustable phase shift on the incident signal.Furthermore,we reshape the complex-valued covariance matrix and reflection matrix to a vector for the ease of calculating partial derivatives to find the search direction.Then,the quasi-Newton updates and modified Broyden-Fletcher-Goldfarb-Shano(BFGS)method in the complex domain form are used to find the local minimum.Finally,numerical simulation results demonstrate that our proposed IRS-aided system using the algorithm performs better than the state-of-the-art and the conventional communication systems.

基于对偶相切和Lagrange插值的高压光伏MPPT算法

为了提高高压光伏发电系统中最大功率点跟踪(MPPT)的速度、精度以及跟踪到最大功率点(MPP)后运行的平稳度,提出一种对偶相切与Lagrange插值相结合的改进MPPT算法。该算法利用对偶切线法的快速性和Lagrange插值法的近似拟合作用,迅速将参考电压定位到最大功率点附近,最后以极小的步长运行到MPP。仿真结果表明:所提出的改进MPPT算法与传统扰动观测法相比,能够更快速、精准地追踪到最大功率点,并以更小的波动持续地输出最大功率,有效地减少了光伏发电系统的功率损失,提高了太阳能的利用率。