Periodic Solutions and Approximate Symmetries

Assuming that a dynamical system (DS) has a periodic solutionwith period T in the neighborhood of an equilibrium, a newDS can be obtained which governs approximately the motion oftrajectories in the neighborhood of the periodic orbit. Approximatesymmetries and first integrals have been discussed for this DS. Theresults have been demonstrated on the Helmholtz oscillator.     

Continuation of Approximate Transformation Groups via Multiple Time Scales Method

Recently, the theory of approximate symmetries was developedfor tackling differential equations with a small parameter. This theoryfurnishes us with a tool, e.g. for constructing approximate groupinvariant solutions. Usually, these solutions are determined by powerseries in the small parameter and hence they are well defined only in asmall region of independent variables. In this paper, we modify theapproximate symmetry analysis by combining it with the multiple timescales method. In this way, we can extend the domain of definition ofapproximate symmetries of differential equations with a small parameterand of their invariant solutions. The method is illustrated by the vander Pol equation. It is shown that, in this example, our approachprovides a group theoretical background of ad hoc methods widelyused in perturbation techniques.     

Approximate Solution of an Axisymmetric Contact Problem with Allowance for Tangential Displacements on the Contact Surface

A structurally nonlinear contact problem of a punch shaped like a paraboloid of revolution is studied. An equation for the contactpressure density is derived with allowance for the radial tangential displacements of the boundary points of an elastic halfspace. A method for constructing a closedform approximate solution is proposed. The effect of the tangential displacements on the main contact parameters is discussed.     

非完整系统的Lie对称性与Noether对称性

研究非完整系统的Lie对称性与Noether对称性及其间的关系,具体研究了Chetaev型变量质量非完整系统和非Chetaev型非完整系统的Lie对称性与Noether对称性。给出Lie对称性导致Noether对称性及Noether对称性导致Lie对称性的条件。     

一类平面Hamilton系统的大范围非线性化近似方法

在非线性动力系统的研究已经进入了占主导地位的时期,对其提出大范围的非线性化近似方法具有特别重要的意义.在本文中,我们主要对于一类典型的Hamilton系统,根据等势线有两个,或者三个交点的不同情形,给出7种不同的大范围最低次非线性化近似系统,并通过积分近似系统给出近似解(轨道).结果表明,近似椭圆周期轨道可通过线性化近似系统得到,而同(异)宿轨道则可通过2、3次非线性化近似系统得到.最后,将近似方法应用于一个具体Hamilton系统的分析.     

辛体系下含对称破缺因素动力学系统的近似守恒律

自冯康先生创立Hamilton系统辛几何算法以来,诸如辛结构和能量守恒等守恒律逐渐成为动力学系统数值分析方法有效性的检验标准之一。然而,诸如阻尼耗散、外部激励与控制和变参数等对称破缺因素是实际力学系统本质特征,影响着系统的对称性与守恒量。因此,本文在辛体系下讨论含有对称破缺因素的动力学系统的近似守恒律。针对有限维随机激励Hamilton系统,讨论其辛结构;针对无限维非保守动力学系统、无限维变参数动力学系统、Hamilton函数时空依赖的无限维动力学系统和无限维随机激励动力学系统,重点讨论了对称破缺因素对系统局部动量耗散的影响。上述结果为含有对称破缺因素的动力学系统的辛分析方法奠定数学基础。     

Complex Oscillations Revisited

The problem of applying some approaches to approximate solutions and the asymmetry of closed phase trajectories in some domain of parameters are examined__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 2, pp. 81–90, February 2005.     

Analytical solutions of the boundary-value problem of nonstationary flow of viscoplastic medium between two plates

Summary Nonstationary flow of a viscoplastic medium between two parallel plates is considered for the case of a varying pressure gradient. The problem is reduced to the Stephan problem, with the condition on the boundary separating the flow domain from the quasi-rigid domain. Four multiparameter families of exact solutions are found. The first family describes the flow decelerations up to a full stop. The second family determines the development of the flow from the state of rest as the pressure gradient increases. The third family describes the development of the flow for the case where (1) the pressure gradient is constant and exceeds the threshold value related to the yield stress, (2) the upper plate does not move, and (3) the lower plate moves with a constant acceleration. Finally, the fourth family determines the flow retardation, when the pressure gradient is constant and is less than the threshold value. The decrease in the flow of the viscoplastic medium can be achieved for certain values of parameters by increasing the quasi-rigid domain, whereas the viscoplastic flow remains unchanged. Received 7 October 1998; accepted for publication 8 April 1999     

Approximate Group Analysis and Multiple Time Scales Method for the Approximate Boussinesq Equation

This paper is devoted to investigation of the approximate Boussinesq equation by methods of the approximate symmetry analysis of partial differential equations with a small parameter developed by Baikov, Gazizov and Ibragimov. We combine these methods with the method of multiple time scales to extend the domain of definition of approximate group invariant solutions of the approximate Boussinesq equation.     

Solvability on boundary-value problems of elasticity of three-dimensional quasicrystals

Weak solution (or generalized solution) for the boundary-value problems of partial differential equations of elasticity of 3D (three-dimensional) quasicrystals is given, in which the matrix expression is used. In terms of Korn inequality and theory of function space,we prove the uniqueness of the weak solution.This gives an extension of existence theorem of solution for classical elasticity to that of quasicrystals,and develops the weak solution theory of elasticity of 2D quasicrystals given by the second author of the paper and his students.     

LIE SYMMETRIES AND CONSERVED QUANTITY OF A BIPED ROBOT

I. INTRODUCTION It is well known there are close relationships between the symmetries and conservation laws inmechanical systems. The symmetric principles are among the key issues in mechanics. Two e?ectivemethods of studying the symmetries and conservation laws of mechanical are Noether’s method[1] andLie’s method. The approach to Lie symmetries was reported in the 19th century, but no applicationin mechanics appeared until 1979[2]. In recent years, studies of Lie’s method have be…     

考虑恒载效应的拱形梁静力近似解

应用虚功原理,推导了考虑恒载效应影响时拱形梁在活载作用下的非线性微分方程,得到了方程的近似闭合解。根据方程的解,讨论了恒载大小及结构自身刚度(矢高、跨度、惯性矩及惯性半径等)不同因素在考虑恒载效应时对拱形梁静力特性的影响。通过与Takabatake得到的直梁解析解结果及作者在其他文献提出的有限元方法对拱形梁分析结果的比较,验证了本文非线性微分方程及其求解公式。结果表明,本文给出的非线性微分方程对于拱形梁和直线梁具有通用性,初始恒载的存在减小了拱形梁在活载作用下的静力反应,这种影响与恒载的大小及结构自身的刚度有关,对轻型结构的设计提出了一些建议。     

Equivalence Classes and Symmetries of the Variable Coefficient Kadomtsev—Petviashvili Equation

We classify the variable coefficient Kadomtsev—Petviashvili (VCKP) equation into equivalence classes under the group of local point transformations, leaving the equation form invariant but changing the coefficient functions. We list the representatives of all equivalence classes with the corresponding transformations. Then, we obtain the symmetry group of the VCKP equation and in particular discuss how to use these transformations to classify low-dimensional symmetry algebras in the generic case. We conclude with a discussion of the implications of the present article.     

Symmetries and Conserved Quantities of Stochastic Dynamical Control Systems

A new definition is given for both exact and quasi symmetries of Itô and Stratonovich dynamical control systems. Determining systems of symmetries for these systems have been obtained and their relation is discussed. It is shown that conserved quantities can be found from both exact and quasi symmetries of stochastic dynamical control systems, which includes Hamiltonian control systems as a special case. Systems which can be controlled via conserved quantities have been investigated. Results have been applied to the control of an N-species stochastic Lotka—Volterra system.     

Approximate sampling theorem for bivariate continuous function

An approximate solution of the refinement equation was given by its mask, and the approximate sampling theorem for bivariate continuous function was proved by applying the approximate solution . The approximate sampling function defined uniquely by the mask of the refinement equation is the approximate solution of the equation , a piece-wise linear function , and posseses an explicit computation formula . Therefore the mask of the refinement equation is selected according to one' s requirement, so that one may controll the decay speed of the approximate sampling function .     

Modeling of Nonlinear Antiplane Strain of a Cylindrical Body

Antiplane strain of an elastic cylindrical body is studied with allowance for geometrical and physical nonlinearities and potential forces. The nonlinear boundary-value problem for two independent strains is solved. An analytical solution and the corresponding load are obtained for the Rivlin-Saunders quadratic elastic potential, which models finite elastic strains. The problem for displacements specified on the boundary is solved. The case of weak physical nonlinearity is considered.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 98–108, July– August, 2005.     

包含伺服约束的非完整系统的对称性摄动与绝热不变量

建立包含伺服约束系统的运动微分方程,基于包含伺服约束系统对称性与不变量理论研究包含伺服约束系统的对称性摄动与绝热不变量 问题,提出了包含伺服约束系统的高阶绝热不变量概念,证明了精确不变量与绝热不变量存在的条件及形式,并举例说明结果的应用。     

Noether-Type Symmetries and Conservation Laws Via Partial Lagrangians

We show how one can construct conservation laws of Euler-Lagrange-type equations via Noether-type symmetry operators associated with what we term partial Lagrangians. This is even in the case when a system does not directly have a usual Lagrangian, e.g. scalar evolution equations. These Noether-type symmetry operators do not form a Lie algebra in general. We specify the conditions under which they do form an algebra. Furthermore, the conditions under which they are symmetries of the Euler-Lagrange-type equations are derived. Examples are given including those that admit a standard Lagrangian such as the Maxwellian tail equation, and equations that do not such as the heat and nonlinear heat equations. We also obtain new conservation laws from Noether-type symmetry operators for a class of nonlinear heat equations in more than two independent variables.     

Application of Symmetries to Central Force Problems

We determine the Noether point symmetries associated with theusual Lagrangian of the differential equation of the orbit of the twobody problem. This gives rise to three definite natural forms (apartfrom the linear form) when the Lagrangian admits three Noether pointsymmetries which enables the solution of the orbit equation in terms ofelementary functions. The other forms for which the orbit equation hastwo point symmetries that arise in the Lie classification do not occurfor the usual Lagrangian. For central force problems we obtain inaddition to the six general central forces found by Broucke [8], newforce laws which lead to integrability in terms of known functions. Thisis achieved by the two Noether cases and further cases by use ofequivalence transformations of the orbit differential equation.Moreover, we give an extension of what is sometimes referred to asNewton's theorem of revolving orbits. We also make use of the Lieclassification to provide two new cases of integrable (in terms of knownfunctions) orbit differential equations and new force laws.