Unconventional Hamilton-type variational principles for analytical mechanics

According to the basic idea of classical yin-yang complementarity and modern dual-complementarity, in a simple and unified new way proposed by Luo, the un-conventional Hamilton-type variational principles of holonomic conservative system in analytical mechanics can be established systematically. This unconventional Hamilton-type variational principle can fully characterize the initial-value problem of analytical mechanics, so that it is an important innovation for the Hamilton-type variational principle. In this paper, an important integral relation is given, which can be considered as the expression of the generalized principle of virtual work for analytical mechanics in mechanics. Based on this relation, it is possible not only to obtain the principle of virtual work of holonomic conservative system in analytical mechanics, but also to derive systematically the complementary functionals for three-field and two-field unconventional variational principles, and the functional for the one-field one by the generalized Legendre transformation given in this paper. Further, with this new approach, the intrinsic relationship among various principles can be explained clearly. Meanwhile, the unconventional Hamilton-type variational principles of nonholonomic conservative system in analytical mechanics can also be established systematically in this paper.     

Some basic principles for the linear theory of piezoelectric micropolar elastodynamics

According to the basic idea of classical yin-yang complementarity and modern dual-complementarity,in a simple and unified way proposed by Luo,some basic principles in the linear theory of piezoelectric micropolar elastodynamics can be established systematically. In this paper,an important integral relation in terms of convolutions is given,which can be considered as the generalized principle of virtual work in mechanics. Based on this relation,it is not only possible to obtain the principle of virtual work and the reciprocal theorem,but also to systematically derive the complementary functionals for the eleven-field,nine-field and six-field simplified Gurtin-type variational principles and the potential energy-functional for the three-field one in the linear theory of piezoelectric micropolar elastodynamics by the generalized Legendre transformations given in this paper. Furthermore,with this approach,the intrinsic relationships among various principles can be explained clearly.     

Some basic principles in dynamic theory of viscoelastic materials with voids

According to the basic idea of classical yin-yang complementarity and modern dual-complementarity, in a simple and unified way proposed by Luo, some basic principles in the dynamic theory of viscoelastic materials with voids can be estab- lished systematically. In this paper, an important integral relation in terms of con- volutions is given, which can be considered as the generalized principle of virtual work in mechanics. Based on this relation, it is possible not only to obtain the principle of virtual work and the reciprocal theorem, but also to derive systemati- cally the complementary functionals for the eight-field, six-field, four-field simpli- fied Gurtin-type variational principles and the potential energy-functional for the two-field one in the dynamic theory of viscoelastic materials with voids by the generalized Legendre transformations given in this paper. Furthermore, with this approach, the intrinsic relationship among various principles can be explained clearly.     

Unconventional Hamilton-type variational principle in phase space and symplectic algorithm

By a novel approach proposed by Luo, the unconventional Hamilton-type variational principle in phase space for elastodynamics of multidegree-of-freedom system is established in this paper. It not only can fully characterize the initial-value problem of this dynamic, but also has a natural symplectic structure. Based on this variational principle, a symplectic algorithm which is called a symplectic time-subdomain method is proposed. A non-difference scheme is constructed by applying Lagrange interpolation polynomial to the time subdomain. Furthermore, it is also proved that the presented symplectic algorithm is an unconditionally stable one. From the results of the two numerical examples of different types, it can be seen that the accuracy and the computational efficiency of the new method excel obviously those of widely used Wilson-θ and Newmark-β methods. Therefore, this new algorithm is a highly efficient one with better computational performance.     

Variational correctness and Timoshenko beam finite element elastodynamics

The finite element discretisation of the two-noded Timoshenko beam element for elastodynamics offers very interesting insights into the error analysis aspects of the formulation. In this paper, the relatively different order of convergence of the two spectra of the Timoshenko beam theory, and the extra-variational aspect of the use of reduced integration to free the element of locking, are investigated. The correct variational basis for finite element analysis of elastodynamic problems is presumed to originate from the principle of virtual work, with a simultaneous consideration of errors in both displacement and strains. A variationally correct element would lock; to make an element free of locking, some degree of variational incorrectness must be brought in. The present paper also demonstrates that reduced integration violates the virtual work principle which in turn causes the loss of boundedness of the finite element eigenvalues with the exact solution.     

精确Cosserat弹性杆动力学的分析力学方法

以脱氧核糖核酸和工程中的细长结构为背景, 大变形大范围运动的弹性杆动力学受到关注. 将分析力学方法运用到精确Cosserat弹性杆动力学, 旨在为前者拓展新的应用领域, 为后者提供新的研究方法. 基于平面截面假定, 在弯扭基础上再计及拉压和剪切变形形成精确Cosserat弹性杆模型. 用刚体运动的概念描述弹性杆的变形, 导出弹性杆变形和运动的几何关系; 在定义截面虚位移及其变分法则的基础上, 建立用矢量表达的d’Alembert-Lagrange原理, 在线性本构关系下化作分析力学形式, 并导出Lagrange方程和Nielsen方程, 定义正则变量后化作Hamilton正则方程; 对于只在端部受力的弹性杆静力学, 导出了将守恒量预先嵌入的Lagrange方程, 并讨论了其首次积分. 从弹性杆的d’Alembert-Lagrange原理导出积分变分原理, 在线性本构关系下化作Hamilton原理. 形成的分析力学方法使弹性杆的全部动力学方程具有统一的形式, 为弹性杆动力学的对称性和守恒量的研究及其数值计算铺平道路. 关键词： 精确Cosserat弹性杆 分析动力学方法 变分原理 Lagrange方程     

Hamilton formalism and variational principle construction

It is widely accepted that a variational principle cannot be constructed for an arbitrary differential equation; a rigorous mathematical condition shows which equations can have a variational formulation. On the other hand, the importance for variational principles in various fields of physics resulted in several methods to circumvent this condition and to construct another type of variational principles for any differential equation. In this paper the common origin of the considered methods is investigated, and a generalized Hamiltonian formalism is formulated. Additionally, constructive algorithms are given by different methods to construct variational principles. Simple examples are presented to make construction methods more transparent: several Lagrangians are constructed for the different forms of the Maxwell equations and for the extended heat conduction equation.     

凑合反推法──流体力学变分原理建立的一条新途径

凑合反推法是刘商联系统方法[1]的进一步发展，应用这种方法可以方便地构造各种亚广义变分原理及广义变分原理，并可以消除临界变分现象．对于任何二维守恒型流体力学方程，作者推导得到了其广义变分通用公式．几个实例证明这种方法是有效的、简单的，并具有普遍的意义．     

事件空间中力学系统的微分变分原理

研究事件空间中力学系统的微分变分原理.基于D'Alembert原理，建立了事件空间中力学系统的D'Alembert-Lagrange原理、Jourdain原理、Gauss原理和万有D'Alembert原理，给出了这些原理的Euler-Lagrange参数形式、Nielsen参数形式和Appell参数形式，并导出了万有D'Alembert原理的Mangeron-Deleanu参数形式. 关键词： 分析力学 事件空间 微分变分原理     

不同积分变分原理的统一

依据定量因果原理的数学表示，统一地导出了Lagrange量中含坐标关于时间一阶、二阶导数 的积分型的Hamilton原理、Voss原理、Hlder原理和Maupertuis-Lagrange原理等，给出了 这些原理的本质联系和统一描述.得出f₀=0并不是通常的保持Euler-Lagrange方 程不 变的结果，而是满足定量因果原理的结果.还得出Lagrange量的所有的积分型变分原理等价 地对应于两类满足定量因果原理的不变形式.同时发现所有积分型变分原理的运动方程都是E uler-Lagrange 方程，但不同条件的变分原理所对应的不同群Ｇ作用下的守恒量是不同 的.从而可对过去众多零散的积分型变分原理有一个系统和深入的理解，并使这些变分原理 自然地成为定量因果原理的推论. 关键词： 变分原理 因果原理 运动方程 对称性     

Kirchhoff弹性杆动力学建模的分析力学方法

以杆的横截面为研究对象，讨论了其自由度，给出了截面虚位移定义，并定义变分和偏微分运算对独立坐标服从交换关系. 给出了曲面约束的基本假设，讨论了约束对截面自由度的影响以及加在虚位移上的限制方程. 从D'Alembert原理出发结合虚功原理，建立了弹性杆动力学的D'Alembert-Lagrange原理，当杆的材料服从线性本构关系时，化作Euler-Lagrange形式、Nielsen形式和Appell形式. 由此导出了Kirchhoff方程以及Lagrange方程、Nielsen方程和Appell方程，得到 关键词： 超细长弹性杆 分析力学方法 Kirchhoff动力学比拟 变分原理     

Some variational principles in electroelastic media under finite deformation

It is proposed that the universal thermodynamic energy variational principle is included in the first law of thermodynamics. Some variational principles in the electroelastic media under finite deformation are derived from this universal thermodynamic variational principle. It is suggested that in the general electroelastic analysis the environment should be considered together with the discussed electroelastic medium. For the variational principle of nonlinear electroelastic media the variation of the electric potential is coupled with the virtual displacement, and the variation of the initial volume should be considered. The Maxwell stress in the initial configuration is naturally derived from this variational principle and it is unique in the second order precision. Supported by the National Natural Science Foundation of China (Grant No. 10472069)     

超细长弹性杆动力学的Gauss原理

研究基于Gauss 变分的超细长弹性杆动力学建模的分析力学方法.分别在弧坐标和时间的广义加速度空间定义虚位移,给出了非完整约束加在虚位移上的限制方程;建立了弹性杆动力学的Gauss原理,由此导出Kirchhoff方程、Lagrange方程、Nielsen方程以及Appell方程;对于受有非完整约束的弹性杆,导出了带乘子的Lagrange方程;建立了弹性杆截面动力学的Gauss最小拘束原理并说明其物理意义. 关键词： 超细长弹性杆动力学 分析力学 Gauss变分 最小拘束原理     

二级树状管路的优化分析

本文讨论了Y型二级管路分别在占地面积、体积以及占地面积、表面积约束下的最优管径比、最优管长比、最优分岔角度。优化方法采用拉格朗日乘子法和虚功原理方法。通过分析发现;最优树状管路的结构参数随约束条件和流动状态的变化而变化.这意味着对层流流动和湍流流动的情况,树状结构最优参数会表现出差异。本文的忧化结果为不同情况下的树状结构设计提供了基本参考。     

Discrete variational principle and first integrals for Lagrange--Maxwell mechanico-electrical systems

This paper presents a discrete variational principle and a method to build first-integrals for finite dimensional Lagrange--Maxwell mechanico-electrical systems with nonconservative forces and a dissipation function. The discrete variational principle and the corresponding Euler--Lagrange equations are derived from a discrete action associated to these systems. The first-integrals are obtained by introducing the infinitesimal transformation with respect to the generalized coordinates and electric quantities of the systems. This work also extends discrete Noether symmetries to mechanico-electrical dynamical systems. A practical example is presented to illustrate the results.     

Dynamical and thermodynamical stability of two-dimensional flows: variational principles and relaxation equations

We review and connect different variational principles that have been proposed to settle the dynamical and thermodynamical stability of two-dimensional incompressible and inviscid flows governed by the 2D Euler equation. These variational principles involve functionals of a very wide class that go beyond the usual Boltzmann functional. We provide relaxation equations that can be used as numerical algorithms to solve these optimization problems. These relaxation equations have the form of nonlinear mean field Fokker-Planck equations associated with generalized “entropic” functionals [P.H. Chavanis, Eur. Phys. J. B 62, 179 (2008)].     

三类大气基本波动的广义变分原理

半反推法是He为了寻求数学物理问题的变分原理而提出的, 可避免由拉氏乘子法引起的临界变分现象. 应用半反推法分别获得了动力气象中Rossby波、大气声波和重力外波等三类基本大气波动的广义变分原理, 并验证了它们的正确性.     

广义Birkhoff系统的时间积分定理

研究了广义Birkhoff系统的时间积分定理.给出系统的时间积分等式,并由此等式导出类能量方程、类维里定理、一个积分变分原理和一个微分变分原理. 关键词： 广义Birkhoff系统 时间积分定理 类能量方程 变分原理     

Boussinesq方程组的广义变分原理

半反推法是何吉欢为了寻求物理问题的变分原理而提出的,可避免由拉氏乘子法引起的临界变分现象. 应用半反推法分别获得了描述水波运动的两类Boussinesq方程组的一族广义变分原理,并验证了它们的正确性. 关键词： 半反推法 广义变分原理 Boussinesq方程组