正则变换下的哈密顿原理

正则变换是相空间中一类变量变换.2n 个正则共轭变量 qi 和 pi 变换成2n 个新变量Q_i=Q_i(q,p,t),P_i=P_i(q,p,t), (1)若在此变换下系统运动方程的正则形式不变,则称此变换为正则变换.设 H(q,p,t)和 H′(Q,P,t)分别是 ...     

变量的两种独立性和哈密顿原理的无穷多样性

变量的两种独立性和哈密顿原理的无穷多样性朱如曾，向程（中国科学院力学研究所，北京１０００８０）最近《力学与实践》发表了“对广义动量的探讨”一文￣［１２］，该文批评了９本力学著作￣［１－２，４－１０］的有关论述￣１）．如果文［１２］的批评是正确的，那么...     

Hamilton原理四种表达形式的异同

Ｈａｍｉｌｔｏｎ原理四种表达形式的异同陈滨（北京大学力学系北京１００８７１）１．引言Ｈａｍｉｌｔｏｎ原理是经典力学中的重要原理。它是以泛函（Ｈａｍｉｌｔｏｎ作用量）驻值的变分形式给出的力学系统动力学原理,并可以作为整个力学系统动力学研究的基础,Ｈａｍ...     

变质量系统哈密顿正则方程和哈密顿原理

从理想完整约束变质量系统的拉格朗日方程出发,导出了这种系统的哈密顿正则方程．从理想约束系统动力学普遍方程出发,导出了理想约束变质量系统哈密顿原理．     

基于非传统哈密顿变分原理的高阶辛算法

给出了非传统哈密顿变分原理的一种简化形式,并在此基础上利用拉格朗日多项式近似位移和动量,采用高斯积分法对时间积分,建立了针对动力学初值问题的一类高阶辛算法。在建立高阶辛算法的过程中,本文方法与基于传统哈密顿变分原理的辛算法不同,无需由端值问题向初值问题转换,因此更加简捷有效。此外,给出了线性动力问题中本文算法保辛性的证明。当位移、动量的插值次数和高斯积分点个数均为m时,本文算法是具有2m阶精度的辛算法,且是线性无条件稳定的。通过数值算例结果表明,本文算法与辛算法性质吻合,并且计算效率比同阶辛龙格库塔法提高了约50%。     

基于哈密顿动力系统新变分原理的保辛算法之一：变分原理和算法构造

提出了哈密顿动力系统的一个新变分原理,并基于此变分原理构造了四类保辛算法。通过新的变分原理定义修正作用量,然后将位移和动量采用拉格朗日多项式近似,并采用高斯积分对时间近似积分得到近似的修正作用量。在修正作用量的基础上,通过选择时间步两端不同的位移或动量作为独立变量,可构造四种不同类型的保辛算法。     

基于哈密顿动力系统新变分原理的保辛算法之三：数值算例

文献[1,2]给出了四种不同类型的求解哈密顿动力系统的数值方法,并证明了它们的保辛特性。本文将讨论这四类算法的具体数值性能,包括算法的线性稳定性,精度和效率等。     

基于哈密顿动力系统新变分原理的保辛算法之二：算法保辛性质证明

文献[1]给出了哈密顿系统的一个新的变分原理,并基于此变分原理,通过选择一个时间步长两端不同广义位移或广义动量为独立变量,给出了四种不同类型的求解哈密顿动力系统的数值方法。本文将分别证明这四类数值方法都是保辛的数值方法。     

爆轰等容终点的四组理论解

用Hamilton原理描述爆轰产物粒子运动并最终耗散为热的过程,证明爆轰过程反应区有四组理论解可以达到等容终点,即反应区的多变量解、等温过程解、等压过程解、等容过程解;修正了经典的ZND模型由于忽略输运效应而认为爆轰过程不可能达到等容终点的结论。本文结果与实验结果较为一致。     

New way to construct high order Hamiltonian variational integrators

This paper develops a new approach to construct variational integrators. A simplified unconventional Hamilton’s variational principle corresponding to initial value problems is proposed, which is convenient for applications. The displacement and momentum are approximated with the same Lagrange interpolation. After the numerical integration and variational operation, the original problems are expressed as algebraic equations with the displacement and momentum at the interpolation points as unknown variables. Some particular variational integrators are derived. An optimal scheme of choosing initial values for the Newton-Raphson method is presented for the nonlinear dynamic system. In addition, specific examples show that the proposed integrators are symplectic when the interpolation point coincides with the numerical integration point, and both are Gaussian quadrature points. Meanwhile, compared with the same order symplectic Runge-Kutta methods, although the accuracy of the two methods is almost the same, the proposed integrators are much simpler and less computationally expensive.     

Hamilton's Principle and Rankine–Hugoniot Conditions for General Motions of Mixtures

In previous papers [1, 2], we presented hyperbolic governing equations and jump conditions for barotropic fluid mixtures. Now we extend our results to the most general case of twocomponent media. We obtain governing equations for each component. This is not a system of conservation laws. Nevertheless, using Hamilton's principle we are able to obtain a complete set of Rankine–Hugoniot conditions. For the twofluid case, the jump relations do not involve the conservation of the total momentum and the total energy.Sommario. In precedenti lavori [1, 2] sono state dedotte equazioni di governo iperboliche e condizioni di salto per miscele fluide barotropiche. I risultati sono estesi al caso più generale di mezzi a due componenti, ottenendo le equazioni di governo per ciascun componente. Questo sistema non è derivabile dalle leggi di conservazione. Nondimeno, usando il principio di Hamilton è possibile ottenere un insieme completo di condizioni di Rankine–Hugoniot. Nel caso dei due fluidi, le condizioni di salto non coinvolgono la conservazione del momento e dell'energia totali.     

一个适用于非线性板壳问题的广义变分原理

本文在现时拉格朗日(UL)描述下推导了适用于有限变形及率敏感非弹性材料的广义变分原理。以位移率和应变率为独立变量使它较适于在非弹性材料的板壳问题中应用。并推导了两种率敏感塑性模型(Bodner和Perzyna)的统一的计算增量表达形式。     

分析力学初值问题的一种变分原理形式

明确了分析力学初值问题的控制方程,按照广义力和广义位移之间的对应关系,将 各控制方程卷乘上相应的虚量,代数相加,进而在 原空间中建立了分析力学初值问题的一种变分原理形式,即建立了分析力学初值问题的卷积 型变分原理和卷积型广义变分原理. 推导了分析力学初值问题卷积型变分原理和卷积型广义 变分原理的驻值条件. 在建立分析力学初值问题的一种变分原理形式的同时, 将变积方法推广为卷变积方法.     

THE STATIONARY VALUE PROPERTY OF HAMILTON'S PRINCIPLE IN NON-HOLONOMIC SYSTEMS

I.IntroductionHertzintroducedtheconceptofnon-holonomics}stems(NHS)in1894ti].HeconsideredthatHamiltoll'sprinciple\"asnotsatisfiedforNHSandinparticulardidnotyieldLagrangeequationswithundeterminedmultipliers.Forthisfeason.Hblderl=],Suslovi3]andVoronetzl']dev…     

Hyperbolic Models of Homogeneous Two-Fluid Mixtures

One derives the governing equations and the Rankine–Hugoniot conditions for a mixture of two miscible fluids using an extended form of Hamilton's principle of least action. The Lagrangian is constructed as the difference between the kinetic energy and a potential depending on the relative velocity of components. To obtain the governing equations and the jump conditions one uses two reference frames related with the Lagrangian coordinates of each component. Under some hypotheses on flow properties one proves the hyperbolicity of the governing system for small relative velocity of phases.     

耦合热弹性接触问题的变分原理

本文给出了考虑库伦摩擦力的热弹性接触问题的一个变分原理，该变分原理在约束条件Ｐｎ≥０和－μＰｎ≤Ｐｔ≤μＰｎ下，在接触边界上自动给出导热条件及剩余互补条件等。从它出发将接触弹性体离散后可直接进行二次规划求解。文中特地引进了因子β，它计及了接触问题中的热量散失和功率损耗     

Two methods of solving dynamic response by discretization of Gurtin variational principle

Based on the Gurtin variational principle and the finite element method two recurrence formulae of solving dynamic response are derived in this paper. The first recurrence formula which is conditionally stable is obtained by adopting “dual temporal finite element” to discretize the Gurtin variational principle and then making use of convolution, while the second, an unconditionally stable one derived by discritizing the variational principle in each time step and taking approprlate integration parameter θ. The paper had been accepted by the XVIth International Congress of IUTAM, Lyngby, Denmark, August, 1984.     

A variational principle for fluid mechanics

Summary A variational principle for fluid mechanics is derived without calling for any additional postulates in any ad hoc way. In the principle derived here, the Lagrangian is essentially the sum of kinetic and heat energy transferred to the fluid, less the sum of its internal and potential energy, less the work done on its exterior (similar to the enthalpy concept), rather than the difference between only kinetic energy and internal energy, as obtained previously by Seliger and Whitham [1] for a more restricted mode of variation.