基于高斯原理的Cosserat弹性杆动力学模型

在动力学普遍原理中, 高斯最小拘束原理的特点是可通过寻求函数极值的变分方法直接得出运动规律, 而无须建立动力学微分方程. Kirchhoff动力学比拟方法以刚性截面的姿态表述弹性细杆的几何形态, 并发展为以弧坐标s和时间t为自变量的弹性杆分析力学. 由于截面姿态的局部微小改变沿弧坐标的积累不受限制, Kirchhoff模型适合描述弹性杆的超大变形. Cosserat弹性杆模型考虑了Kirchhoff模型忽略的截面剪切变形、中心线伸缩变形和分布力等因素, 是更符合实际弹性杆的动力学模型. 建立了基于高斯原理的Cosserat弹性杆的分析力学模型, 导出拘束函数的普遍形式, 以平面运动为例进行讨论. 关于弹性杆空间不可自相侵占的特殊问题, 给出相应的约束条件对可能运动施加限制, 以避免自相侵占情况发生.     

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     

弹性细杆平衡的动态稳定性

从动力学观点讨论弹性细杆的平衡稳定性问题.建立了弹性杆的动力学方程，导出了杆的弯扭度与截面角速度之间的运动学关系式.对具有弧坐标s和时间t双重自变量的离散动力系统扩充了Lyapunov 稳定性定义.以具有初扭率的非圆截面直杆的平衡稳定性为例，应用一次近似方法证明了当静力学意义下的稳定性条件得到满足时，动力学意义下的稳定性条件必同时满足. 关键词： 弹性杆动力学方程 Kirchhoff理论 Lyapunov稳定性     

超细长弹性杆的Mei对称性及其Noether守恒量

基于Kirchhoff的动力学比拟,用动力学的概念和方法研究圆截面弹性杆的Hamilton函数和方程,并给出弹性杆的Mei对称性定义和定理以及定理的证明,最后给出弹性杆动力学系统的Mei对称性导致Noether守恒量的条件及定理,并给出算例. 关键词： 超细长弹性杆 Mei对称性 Noether守恒量     

准坐标下广义非保守系统Lagrange方程的积分因子与守恒定理

用积分因子方法研究准坐标下广义非保守系统Lagrange方程的守恒定理.列写系统的运动微分方程，给出它的积分因子的定义.研究守恒量存在的必要条件.建立系统的守恒定理及其逆定理，并举例说明结果的应用. 关键词： 准坐标 Lagrange方程 积分因子 守恒量     

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     

光孤子约束系统的量子场论

光孤子系统可用奇异Lagrange量描述,系统含Dirac约束.通常按对应原理写出系统对易关系和量子运动方程时,未计及约束.文中对该系统进行严格的Dirac括号量子化,给出了系统的对易关系和量子运动方程,还对系统进行了路径积分量子化,并根据量子水平的Noether定理,导出了系统在时空平移变换不变性下的量子能量和动量守恒.系统还具有相位变换下的不变性,相应导出了系统的粒子数守恒.     

弹性压扭直杆的Greenhill公式对精确模型的推广

将圆截面Kirchhoff弹性压扭直杆的Greenhill公式推广到精确模型.基于平面截面假定,在弯扭的基础上增加了拉压和剪切变形,将弹性杆的位形表达为截面的弧坐标历程.由弹性杆精确模型的平衡微分方程,得到了两端受力螺旋作用时对应于直线平衡状态的特解,导出了线性化扰动方程及其通解,再根据两端为铰支时的边界条件以及积分常数存在非零解的条件导出弹性直杆精确模型的Greenhill公式.结果表明,由力螺旋表示的稳定域为一对称的封闭区域,拉压和剪切对稳定性的影响取决于拉压柔度与剪切柔度之差、抗弯刚度和杆长这三个因素.     

Landau-Lifschitz铁磁方程的Hamilton理论和规范变换

对完全各向同性Heisenberg铁磁链的LandauLifschitz方程的Hamilton理论建立中，Hamilton量的坐标积分和谱参数积分两种表示式不能协调地从单一守恒量导出的问题，利用规范变换完善地解决了.并可推广后处理非各向同性铁磁链的LandauLifschitz方程的Hamilton理论. 关键词： 规范变换 LandauLifschitz方程 守恒量 Hamilton理论     

特定条件下相对动点的动量矩定理的简洁形式及其应用

理论力学教材中,刚体平面运动微分方程是利用质心运动定理结合相对质心的动量矩定理导出的,其对于解决刚体平面运动动力学问题提供了普遍的方法,但在有些情况下,采用其他形式的平面运动方程可以更为简洁.为提高特定条件下应用平面运动微分方程解题的效率,给出了质点系相对动点的动量矩定理的一般形式,结合质心运动定理,得到平面运动微分方程的其他形式.讨论了特殊情况下定理的简化条件及简化形式,举例说明了简化形式的动量矩定理结合质心运动定理在解题中的应用,与一般形式的平面运动微分方程相比,可以使解题过程大为简化.     

Noether symmetry and conserved quantities of the analytical dynamics of a Cosserat thin elastic rod

In this paper, we investigate the Noether symmetry and Noether conservation law of elastic rod dynamics with two independent variables： time t and arc coordinate s. Starting from the Lagrange equations of Cosserat rod dynamics, the criterion of Noether symmetry with Lagrange style for rod dynamics is given and the Noether conserved quantity is obtained. Not only are the conservations of generalized moment and generalized energy obtained, but also some other integrals.     

Jourdain Principle of a Super-Thin Elastic Rod Dynamics

A super thin elastic rod is modeled with a background of DNA super coiling structure, and its dynamics is discussed based on the Jourdain variation. The cross section of the rod is taken as the object of this study and two velocity spaces about are coordinate and the time are obtained respectively. Virtual displacements of the section on the two velocity spaces are defined and can be expressed in terms of Jourdain variation. JourdMn principles of a super thin elastic rod dynamics on arc coordinate and the time velocity space are established, respectively, which show that there are two ways to realize the constraint conditions. If the constitutive relation of the rod is linear, the Jourdain principle takes the Euler-Lagrange form with generalized coordinates. The Kirchhoff equation, Lagrange equation and Appell equation can be derived from the present Jourdain principle. While the rod subjected to a surface constraint, Lagrange equation with undetermined multipliers may be derived.     

On Synge's covariant conservation laws for general relativity

Following Synge, the covariant formulas for the total four-momentum and angular momentum of an isolated physical system in general relativity are derived. These formulas are first obtained in the weak-field approximation, for which they are shown to be expressible in surface integral form, to be unique, and to represent covariantly conserved quantities. The covariant expressions for the general case are then shown to be identical to those for the weak-field case. The uniquely determined and covariantly conserved quantities so obtained are found to agree with the corresponding canonical, noncovariant surface integral expressions.     

Theory of the free full-circle arc

A theoretical investigation of the full-circle arc located between two planes is presented. The circular arc shape is due to an applied magnetic field. The basic equations for conservations of mass, momentum, energy, and charge, as well as Maxwell's equations and the equation of state lead to a coupled set of partial differential equations. By means of Green's formula, this set is transformed into a set of integral equations. Using the analytically known Green's function, the system may be solved by an iteration procedure. For a simplified arc model, the quantities of interest are computed: The temperature distribution, the mass flow field, and the external magnetic field necessary to maintain this arc configuration.     

Conservation laws for a vortex in a compressible nonequilibrium medium

The problem of conservation of magnitudes is considered for a vortex in a relaxing compressible medium. Heat release due to the relaxation of a nonequilibrium medium leads to the propagation of compression waves, which remove material. Traditional integrals of motion are inapplicable in this case. We pro-pose the concept of integral quantity, which is conserved with an arbitrary degree of accuracy despite the fact that waves cross the boundary of the integration domain. Based on this concept, a broad class of conservation laws is derived for axisymmetric disturbances of columnar vortices, including conservation of the circulation and total angular momentum of the vortex. For nonaxisymmetric disturbances, it is shown that the total angular momentum and properly defined energy integral are conserved. Numerical verification of the derived conservation laws is performed and the perspectives for using these conservation laws in numerical simulations are discussed.     

Existential theorem of conserved quantities and its inverse for the dynamics of nonholonomic relativistic systems

We present a general approach to the construction of conservation laws for the dynamics of nonholonomic relativistic systems.Firstly,we give the definition of integrating factors for the differential equations of motion of a mechanical system.Next,the necessary conditions for the existemce of the conserved quantities are studied in detail.Then,we establish the existential theorem for the conserved quantities and its inverse for the equations of motion of a nonholonomic relativistic system.Finally,an exampled is given to illustrate the application of the result.