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.     

具有一个Fermi自由度的量子模型

详细讨论了具有一个Fermi自由度的量子模型 .模型的算子作用量具有定域算子规范对称性 .从一些具体的物理上的要求出发 ,可得到一组关于算子规范势B0 和规范变换算子U的约束条件 .Fermi型算子 ψ的Euler Lagrange运动方程给出通常的Fermi型运动方程 ,而算子规范势B₀ 的Euler Lagrange方程则只是一个约束 ,此约束就是通常Fermi子的正则量子化条件 .     

平面运动n级倒立摆系统建模及统一表示

针对小车平面运动n级倒立摆系统,利用Lagrange方程和刚体力学的一些理论,给出了模型建立过程,最后得到了平面运动n级倒立摆的统一数学模型.     

不可压缩流体的湍流理论

本文从湍流Reynolds平均运动方程和逐阶速度关联的动力学方程出发,引进了准相似性条件和对关联方程中耗散项的假定,利用逐级近似方法发展了湍流理论.作为一级近似的应用例子,我们曾在忽略三阶关联项下,求解了湍流槽流,平面尾流和射流的平均运动方程和二阶关联方程,得到了理论和实验符合的结果. 本文在一级近似的基础上,进一步引进了三阶和四阶关联的方程作为二级近似,并求解了平面湍流尾流.得出的三阶关联的理论计算和现有的实验的比较是令人满意的.理论也给出了四阶关联。它可以用实验来验证.也可以用它求更高阶的关联.     

一个新的量子力学模型

讨论了具有一个Bose自由度的量子力学模型.用传统的方法构造量子力学模型时,需要知道相应的经典模型,并利用经典Poisson括号与正则量子对易子之间的对应关系来得到正则量子化条件.在讨论的量子模型里,不需要首先讨论相应的经典模型.此模型里,Lagrange量具有算子规范不变性,定域的理论中需引入相应的规范势.动力学变量的Euler-Lagrange运动方程就是通常的运动方程,而规范势仅仅给出一个约束,而此约束刚好是正则量子化条件.     

变质量非完整系统打击运动的Kane方程

本文建立变质量非完整系统Routh形式的Kane方程,并由此导出变质量完整系统和非完整系统打击运动的Kane方程,其次指出Lagrange形式的打击运动方程与Kane方程的等价性。最后举例说明新方程的应用。     

受匀速转动约束的变质量完整力学系统的运动微分方程

本文建立受匀速转动约束的变质量完整力学系统的运动微分方程,包括Lagrange形式的方程,Nielsen形式的方程和Appell形式的方程,并举例说明这些新型方程的应用。     

Kahler流形上的Lagrange力学

讨论了Kahler流形上的Lagrange力学,并给出Lagrange算子、Lagrange方程、作用泛函、Hamilton原理和Hamilton方程等复的数学形式.     

求解二维Poisson方程的重心插值配点法

采用重心Lagrange插值配点法计算了二维Poisson方程.采用重心Lagrange插值法构造近似函数,由配点法离散Poisson方程及其边界条件.数值算例表明方法具有理论简单、计算精度高的特点.     

K(a)hler流形上的Lagrange力学

讨论了K(a)hler流形上的Lagrange力学,并给出Lagrange算子、Lagrange方程、作用泛函、Hamilton原理和Hamilton方程等复的数学形式.     

Nonlinear dynamics of a simplified engine-propeller system

This paper presents a procedure for studying dynamical behaviors of a simplified engine-propeller dynamical system consisting of a number of bodies of plane motions. The equation of motion of the complex system is obtained using the Lagrange equation and solved numerically using the 4th order Runge–Kutta method. Various simulations were performed to investigate the transient and steady state behaviors of the multiple body system while taking into consideration the engine pressure pulsations, nonlinear inertia of moving bodies, and nonlinear aerodynamic load. Sub-harmonics and super harmonics in the steady state responses for different power and propeller pitch settings are obtained using the fast Fourier transform. Numerical simulations indicate that the 1.5 order is the dominant order of harmonics in the steady state oscillatory motion of the crankshaft. The findings and procedure presented in the paper are useful to the aerospace industry in certifying reciprocating engines and propellers. The crankshaft oscillatory velocities obtained from the simplified rigid body model are in good agreement with the experimental data for a SAITO-450 engine and a SOLO propeller at a 6″ pitch setting.     

Vibrations of Pipes with Internal Flow and Rigid Body Degrees of Freedom

In the present paper the nonlinear dynamics of elastic pipes conveying fluid at arbitrary flow rates are investigated. The nonlinear equations of motion are derived using a unified form of the Lagrange Equations for non-material volumes formulated by Irschik and Holl [1], see also Chapter 3 of [2]. In a first step cantilevered pipes are considered using elastic degrees of freedom combined with a Ritz-Galerkin Ansatz of arbitrary order for modelling the deformations of the pipes. The Lagrange Equations for non-material volumes include a nonzero surface integral of the kinetic energy due to the moving outlet surface at the end of the pipe. The linear equations of motion obtained from this model are then analytically investigated utilizing the corresponding Eigenvalue problem. The results are visualized in an Argand representation of the corresponding Eigenvalues of the system matrix and compared to existing results obtained by using different formulations, such as the Hamilton Principle for Open-Systems, formulated by Benjamin [4], as demonstrated by Païdoussis [5], see also chapter 3.5 of [6]. In a next step an elastic pipe with a rigid body degree of freedom combined with a Ritz-Galerkin Ansatz is modelled with one supported and one free end. The derivation of the equations of motion is performed by using a floating-frame of reference formulation which leads to a system of nonlinear second order differential equations describing the motion of the pipe. Finally, the stability of the solutions of the equations of motion for varying flow rate is studied numerically. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the construction of a set of stochastic differential equations on the basis of a given integral manifold independent of velocities

We construct the Lagrange equation, Hamilton equation, and Birkhoff equation on the basis of given properties of motion under random perturbations. It is assumed that random perturbation forces belong to the class of Wiener processes and that given properties of motion are independent of velocities. The obtained results are illustrated by an example of motion of an Earth satellite under the action of gravitational and aerodynamic forces.     

Closed Hypersurfaces Driven by Mean Curvature and Inner Pressure

We introduce a hyperbolic equation that describes the motion of closed hypersurfaces in a Riemannian manifold with surface tension and inner pressure as driving forces. In the case of spherical surfaces this equation can be considered as an idealized mathematical model for a moving soap bubble. The equation is derived as an Euler‐Lagrange equation from a suitable action integral. It is a quasi‐linear degenerate hyperbolic PDE of second order that describes the motion of the surfaces extrinsically. Our main results are the solution of the Cauchy problem by means of the Nash‐Moser inverse function theorem, a continuation criterion, and stability estimates. © 2012 Wiley Periodicals, Inc.     

The oscillations of a body with an orbital tethered system

The motion about a centre of mass of a rigid body with a tethered system, designed to launch a re-entry capsule from a circular orbit is considered. In the deployment of the tethered system the direction and value of the tensile strength of the tether vary and, if the point of application of the tensile strength does not coincide with the centre of mass of the body, a moment occurs which leads to oscillations of the body with variable amplitude and frequency. A non-linear equation of the perturbed motion of the body about the centre of mass under the action of the tensile force of the tether and the gravitational moment is derived. Assuming that the change in the value and direction of the tensile force is slow and also that the gravitational moment is small, approximate and exact solutions of the non-linear differential equation of the unperturbed motion are obtained in terms of elementary functions and elliptic Jacobi functions. For perturbed motion, the action integral is expressed in terms of complete elliptic integrals of the first and second kind.     

Spatial chaotic vibrations when there is a periodic change in the position of the centre of mass of a body in the atmosphere

The spatial chaotic motion of a blunt body in the atmosphere when there is a periodic change in the position of the centre of mass is considered. A restoring moment, described by a biharmonic dependence on the spatial angle of attack, a small perturbing moment, due to the periodic change in the position of the centre of mass, and also a small damping moment, acts on the body. The motion when the velocity head remains constant is investigated. When there are no small perturbations, the phase portrait of the system can have points of stable and unstable equilibrium. The behaviour of the system in the neighbourhood of the separatrice is investigated using Mel’nikov's method. An analytic solution of the equation of the body motion along the separatrice is obtained. The criteria for the occurrence of chaos are obtained and the results of numerical modelling, which confirm the correctness of the solutions obtained, are presented.     

On the governing equations of motion of nonholonomic systems on Riemannian manifolds

We propose a geometric approach to formulate the governing equations of motion for a class of nonholonomic systems on Riemannian manifolds. We first present a coordinate-free geometric formulation of the D’Alembert–Lagrange equation. Then by explicating this geometric formulation with respect to an arbitrary frame, we obtain the governing equations of motion in generalized form. The governing equations so obtained directly eliminate the dependent variations without using undetermined multipliers. As examples, we apply the formulation to a rigid body and a system with general first-order nonholonomic constraints; we also demonstrate their equivalences to the known results.     

A Mean-variance Problem in the Constant Elasticity of Variance (CEV) Mo del

In this paper, we focus on a constant elasticity of variance (CEV) model and want to find its optimal strategies for a mean-variance problem under two con-strained controls: reinsurance/new business an...     

变质量可控力学系统的相对论性变分原理与运动方程*

本文同时考虑经典变质量和相对论变质量情况,建立了基本形式、Lagrange形式,Nielsen形式和APPell形式的变质量可控力学系统的相对论性D'Alembert原理,得到了变质量非完整可控力学系统在准坐标下和广义坐标下的相对论性方程、Nielsen方程和APPell方程,并讨论了完整系统、常质量系统的相对论性可控力学系统的运动方程。