五体问题的中心构型及其相对平衡解

N体问题的中心构型非常重要,但它们的分类很复杂.本文讨论了一类菱形五体问题的中心构型及其相对平衡解,证明了菱形五体问题的相对平衡解的存在唯一性.     

一类新的关于(4+4+1)-体共轭套中心构型

本文研究R3中一类(4+4+1)-体中心构型.利用中心构型等价类的性质及代数、分析方法,得到了该构型构成中心构型的充分和必要条件,证明了对任意给定的质量比这类中心构型存在的结论,解决了给定不同质量比范围该类中心构型是否唯一的问题,推广了文[16]的结论.     

带任意质量的平面2N＋1体问题的周期解

本文研究平面2N+1-体问题的周期解.利用Hermite矩阵和循环矩阵的性质,给出了几何中心带一个质量的平面正多边形套问题存在周期解的必要和充分条件,证明了该周期解的存在唯一性,推广了文献[9]的结论.     

对称迭层矩形板的平面应力分析

常用的对称迭层板为各向异性板．根据平面应力问题的基本方程精确地用应力函数解法求得了各向异性板的一般解析解．推导出平面内应力和位移的一般公式,其中积分常数由边界条件来决定．一般解包括三角函数和双曲函数组成的解,它能满足4个边为任意边界条件的问题．还有代数多项式解,它能满足4个角的边界条件．因此一般解可用以求解任意边界条件下的平面应力问题．以4边承受均匀法向和切向载荷以及非均匀法向载荷的对称迭层方板为例,进行了计算和分析．     

二维各向异性压电介质机电耦合场的基本解

本文研究各向异性压电介质的机电耦台问题．应用平面波分解法和留数定理，首次得到了线力和线电荷作用下一般二维各向异性压电介质机电耦合场的基本解．本文的解适用于平面问题、反平面问题以及平面和反平面相互耦合问题．作为特例，文中给出了横观各向同性压电介质的基本解．     

正四面体仿射套型九体中心构型(英文)

本文研究R3中一类九体中心构型,该构型由两个正四面体及几何中心带一个质量构成.利用中心构型等价类的性质及分析方法,得到了该构型构成中心构型的充分和必要条件,并且证明了对任意给定的质量比这类中心构型存在且唯一,推广了文献[16]的结论.     

三个不动中心问题的一个三维解

本文对三个不动中心问题的平面解作进一步发展，通过对三个不动中心在四维空间中的定义得到三个不动中心问题的三维解．     

复合材料桥纤维拔出问题的动态裂纹模型

在一无限的正交各向异性体的弹性平面上,对具有桥纤维平行自由表面的一个内部中央裂纹,进行了弹性分析．提出了复合材料桥纤维拔出的一个动态模型．由于纤维破坏是由最大拉应力支配,纤维断裂并且裂纹扩展将以自相似的方式出现．通过复变函数的方法将所讨论的问题转化为Reimann-Hilbert混合边界值问题的动态模型,呈现一简单的和容易的解．求得了正交异性体中扩展裂纹受运动的阶梯载荷、瞬时脉冲载荷作用下问题的解析解,并利用这一解,通过迭加最终求得该模型的解．     

高超声速飞行器前体压缩性能研究

用数值模拟的方法研究了高超声速飞行器前体不同型面与进气性能的关系,分析了二级压缩构型在不同压缩角组合和来流攻角下的流场品质和产生的压缩效果．研究结果表明,对于高超声速飞行器,采用多级压缩的前体构型可以得到较好的预压缩效果和优良的流场品质;同时,攻角和不同的压缩角组合会对压缩性能产生影响．因此,采用多级压缩的前体构型、优化各级压缩角的组合是决定飞行器前体预压缩性能的重要因素,同时也是开展前体/发动机一体化设计的关键．     

有限高狭长压电体中半无限反平面裂纹分析

利用保角变换和复变函数方法,研究了裂纹面上受反平面剪应力和面内电载荷共同作用下的有限高狭长压电体中半无限裂纹的断裂问题,给出了电不可通边界条件下裂纹尖端场强度因子和机械应变能释放率的解析解．当狭长体高度趋于无限大时,可得到无限大压电体中半无限裂纹的解析解．若不考虑电场作用,所得解可退化为纯弹性材料的已知结果．此外,通过数值算例,分析了裂纹面上受载长度、狭长体高度以及机电载荷对机械应变能释放率的影响规律．     

Some Enumeration Problems on Central Configurations at the Bifurcation Points

In “Counting central configurations at the bifurcation points,” we proposed an algorithm to rigorously count central configurations in some cases that involve one parameter. Here, we improve our algorithm to consider three harder cases: the planar \((3+1)\)-body problem with two equal masses; the planar 4-body problem with two pairs of equal masses which have an axis of symmetry containing one pair of them; the spatial 5-body problem with three equal masses at the vertices of an equilateral triangle and two equal masses on the line passing through the center of the triangle and being perpendicular to the plane containing it.While all three problems have been studied in two parameter cases, numerical observations suggest new results at some points on the bifurcation curves. Applying the improved version of our algorithm, we count at those bifurcation points. As a result, for the \((3+1)\)-body problem, we identify three points on the bifurcation curve where there are 8 central configurations, which adds to the known results of \(8,9,10\) ones. For our 4-body case, at the bifurcation points, there are 3 concave central configurations, which adds to the known results of \(2,4\) ones. For our 5-body case, at the bifurcation point, there is 1 concave central configuration, which adds to the known results of \(0,2\) ones.     

On the central configurations of the planar restricted four-body problem

This article is devoted to answering several questions about the central configurations of the planar (3+1)-body problem. Firstly, we study bifurcations of central configurations, proving the uniqueness of convex central configurations up to symmetry. Secondly, we settle the finiteness problem in the case of two nonzero equal masses. Lastly, we provide all the possibilities for the number of symmetrical central configurations, and discuss their bifurcations and spectral stability. Our proofs are based on applications of rational parametrizations and computer algebra.     

New central configurations for the planar 7-body problem

In this paper we study numerically the existence of families of planar central configurations for the 7-body problem with the following properties: five bodies are on the vertices of a regular pentagon and the other two bodies are symmetrically positioned in the interior of the pentagon.     

Rose solutions with three petals for planar 4-body problems

For planar Newtonian 4-body problems with equal masses, we use variational methods to prove the existence of a non-collision periodic choreography solution such that all bodies move on a rose-type curve with three petals.     

Minimizing configurations and Hamilton-Jacobi equations of homogeneous N-body problems

For N-body problems with homogeneous potentials we define a special class of central configurations related with the reduction of homotheties in the study of homogeneous weak KAM solutions. For potentials in 1/r ^α with α ∈ (0, 2) we prove the existence of homogeneous weak KAM solutions. We show that such solutions are related to viscosity solutions of another Hamilton-Jacobi equation in the sphere of normal configurations. As an application we prove for the Newtonian three-body problem that there are no smooth homogeneous solutions to the critical Hamilton-Jacobi equation.     

Homographic solutions of the n-body problem in E4

We prove the existence of many homographic solutions of the n-body problem in E⁴ by topological methods. Homographic solutions are associated with relative equilibria. Homothetic solutions always give rise to central configurations. In Euclidean space E⁴ central configurations are a proper subset of the relative equilibria for any n ? 3 and for any (m_i)?R₊ⁿ. We compare the existence and classification of homographic solutions of the n-body problem in E³ with the Newtonian potential and that of homographic solutions of the n-body problem in E⁴. Classifying relative equilibria leads to classifying homographic solutions.     

Numerical Computation of Figure-eight Solution for 3-body Problems

The main goal of this paper is to compute the Figure-eight solutions for the planar Newtonian 3-body problem with equal masses by finding the critical points of the functional associated with the motion equations of 3-body in plane R2. The algorithm adopted here is the steepest descent method, which is simple but very valid for our problem.     

三体问题8字形解的数值计算

The main goal of this paper is to compute the Figure-eight solutions for the planar Newtonian 3-body problem with equal masses by finding the critical points of the functional associated with the motion equations of 3-body in plane R2. The algorithm adopted here is the steepest descent method, which is simple but very valid for our problem.     

On the minimizing properties of the 8-shaped solution of the 3-body problem

The starting point of our study was the recent results of Alain Chenciner and Richard Montgomery concerning the discovery of the 8-shaped orbit of the planar 3-body problem with equal masses (in the sequel, we will call it just “the Eight,” [4]). Geometrically this orbit consists of 12 pieces such that each of them minimizes the Lagrangian action between Euler and isosceles configurations of the bodies. Our aim was to understand whether the larger pieces of the Eight are still solutions of some minimizing problem. The paper presents some preliminary analytical and numerical results on the minimizing properties of the Eight. Using the technique of the so-called Jacobi curves, we numerically show that the solution of Chenciner and Montgomery is no longer optimal after 0.52 of its period. Moreover, we find a better solution for the fixed endpoint problem. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 21, Geometric Problems in Control Theory, 2004.     

The central configurations of four masses x, −x, y, −y

The configuration of a homothetic motion in the N-body problem is called a central configuration. In this paper, we prove that there are exactly three planar non-collinear central configurations for masses x, −x, y, −y with x≠y (a parallelogram and two trapezoids) and two planar non-collinear central configurations for masses x, −x, x, −x (two diamonds). Except the case studied here, the only known case where the four-body central configurations with non-vanishing masses can be listed is the case with equal masses (A. Albouy, 1995-1996), which requires the use of a symbolic computation program. Thanks to a lemma used in the proof of our result, we also show that a co-circular four-body central configuration has non-vanishing total mass or vanishing multiplier.