牛顿运动方程组的拟齐次分类

发布时间：2018-04-26 05:02

本文选题：牛顿运动方程组 + 多项式微分系统　；参考：《吉林大学》2017年硕士论文

【摘要】：常微分方程是伴随着微积分发展起来的,其成长于生产实践和数学的发展进程,蕴含着丰富的数学思想方法.它在天体力学和其它力学领域显示出巨大的功能.牛顿通过解微分方程证实了地球绕太阳的运动轨道是一个椭圆;海王星的存在是天文学家先通过微分方程的方法推算出来,然后才实际观测到的.常微分方程的形成和发展与力学、天文学、物理学以及其他科学技术的发展也有着密切的联系.数学的其他分支的新发展,如复变函数、李群、组合拓扑学等,都对常微分方程的发展产生了深刻的影响.目前,常微分方程在所有自然科学领域和众多社会科学领域都有着广泛的应用.可以预测随着社会技术的发展和需求,常微分方程会有更大的发展.常微分方程的发展经历了四个阶段:第一阶段是以求通解为主要内容的经典理论阶段.1690年,Bernoulli James研究了与钟摆运动有关的“等时曲线问题(在相等的时间内,使摆沿着这条曲线作一次完全的振动(不考虑摆所经历的弧长的大小))”.他通过分析建立了常微分方程的模型,并用分离变量法解出了这条摆线的方程.1690年,Bernoul-1i James 提出了“悬链线问题(绳子悬挂于两固定点而形成的曲线(绳子是柔软的但不能是伸长的))”.Bernoulli John和Leibniz用微积分的方法解决了悬链线问题.后来又研究了等角轨线问题,正交轨线问题等等.1691年,Leibniz给出了变量分离法.1694年,他使用了常数变易法把一阶常微分方程化成积分,又发现了方程的一个解族的包络也是解.1695年,Bernoulli John给出著名的Bernoulli方程.Leibniz用变换将其化为线性方程.1715-1718年,Taylor讨论微分方程的奇解、包络和变量代换公式.1734年,Clairaut研究了 Clairaut方程,发现这个方程的通解是直线族,而直线的包络线就是奇解,Clairaut和Euler对奇解进行了全面的研究,给出从微分方程本身求奇解的方法.1734年,Euler给出了恰当方程的定义.他与克莱罗各自找到了方程是恰当方程的条件,并发现若方程是恰当的,则方程是可积的.1739年克莱罗提出了积分因子的概念,Euler确定了可采用积分因子的的方法求解方程.1772年,Laplace将奇解概念推广到高阶方程和三个变量的方程.1774年,Lagrange对奇解和通解的联系作了系统的研究,他给出了一般的方法及奇解是积分曲线族包络的几何解释等等.第二阶段是以定解问题为研究内容的适定性理论阶段.此时期是数学发展史上的一个转变时期,数学分析的基础、群的概念、复变函数的开创等都在这个时期,常微分方程深受这些新概念和新方法的影响,进入了它发展的第二个阶段.这一阶段的主要结果有:19世纪20年代,柯西建立了柯西问题解的存在唯一性定理.1873年,李普希兹提出著名的“李普希兹条件”,对柯西的存在唯一性定理作了改进.1875年和1876年柯西、李普希兹、皮亚拿和比卡先后给出常微分方程的逐次逼近法等等.第三阶段是常微分方程发展的解析理论阶段.这一阶段的主要结果之一是运用幂级数和广义幂级数解法,求出一些重要的二阶线性方程的级数解,并得到极其重要的一些特殊函数,Riemann-Fuchs奇点理论也是这一阶段非常重要的成果.第四阶段是常微分方程的定性理论阶段,庞加莱和李雅普诺夫分别开创了微分方程定性理论和微分方程运动稳定性理论.多项式微分系统是一类简单而又重要的常微分方程,极限环问题的研究在微分方程的定性理论中占有很重要的地位,1900年,Hilbert提出的第十六个问题的后半部分就是讨论平面多项式系统的极限环的最多个数和相对位置.齐次多项式微分系统作为多项式系统中重要的一类.到目前为止,齐次多项式微分系统已有不少的成果,Markus研究了P,Q互质的二次齐次多项式向量场(P,Q)的分类.Algaba得到了齐次多项式微分系统的标准型以及系统有效的不变量理论.Cima得到实数域上四阶二元型的分类定理和代数特征的分类形式.拟齐次多项式微分系统是齐次多项式微分系统的推广.近年来,拟齐次系统受到众多学者的关注,例如:拟齐次分解,拟齐次多项式系统的可积性,中心问题,极限环,标准型等都取得了丰富的成果.2013年,Garcia给出了一种对平面拟齐次多项式系统进行拟齐次分类的算法,并利用该算法得到了平面2次和3次多项式系统的所有拟齐次分类.本文考虑牛顿运动方程组其中f(q1,q2)和g(q1,q2)分别是q1,q2的n阶和m阶多项式.我们首先探讨方程组(1)的一些拟齐次性质,然后给出(1)的拟齐次分类算法,最后利用算法给出当m ≤ n = 4时(1)的拟齐次分类,记权向量ω =(s1,s2,s3,s4,d),拟齐次向量场为(P1,p2,f,g).分类结果列表如下:
[Abstract]:The ordinary differential equation is developed with calculus, which grows in the development of production practice and mathematics. It contains rich mathematical thought methods. It shows great function in the field of celestial mechanics and other mechanics. By solving differential equations, Newton proved that the orbit of the earth's orbit around too Yang is an ellipse; Neptune's existence is an ellipse. The formation and development of ordinary differential equations are closely related to the development of mechanics, astronomy, physics, and other science and technology. The new development of other branches of mathematics, such as complex functions, Li Qun, combinatorial topology, etc., are all very small. The development of the differential equation has a profound influence. At present, the ordinary differential equation is widely used in all the fields of natural science and many fields of social science. It can be predicted that the ordinary differential equation will have more development with the development and demand of social technology. The development of ordinary differential equations has experienced four stages: the first stage is to seek for the development of ordinary differential equations. In the classical theoretical phase of.1690, Bernoulli James studied the "isochronous curve problem" related to the pendulum movement (in equal time, making a pendulum vibrate along this curve (the size of the arc length experienced by the pendulum). "He established a model of ordinary differential equations through analysis and used separation. The equation of variable method solved the equation of the cycloid for.1690 years, and Bernoul-1i James proposed the "catenary problem (the curve of the rope is suspended at the two fixed point (the rope is soft but not elongated)" ".Bernoulli John and Leibniz used the calculus method to solve the suspension chain problem. Later, the problem of isometric trajectories was studied, and the orthogonality was studied. The trajectory problem and so on.1691 years, Leibniz gives the variable separation method.1694, he uses the constant variable method to integrate the first order ordinary differential equation into integral, and finds the envelope of a solution family of the equation is also.1695 year, Bernoulli John gives the famous Bernoulli equation.Leibniz with the transformation to the linear equation.1715-1718 year, Taylor to discuss. On the singular solutions of differential equations, envelopes and variable substitution formulas in.1734 years, Clairaut studies the Clairaut equation. It is found that the general solution of the equation is a straight line, and the envelope of the line is an odd solution. Clairaut and Euler have carried out a comprehensive study of the singular solutions. The method of finding odd solutions from the differential equations is given for.1734 years, and the proper equation is given by Euler. Definition. He and Kleiro each found the condition of the equation as the proper equation, and found that if the equation is appropriate, then the equation is the integrable.1739 year Kleiro put forward the concept of integral factor, Euler determines the method of integrating the integral factor to solve the equation.1772 year, Laplace extends the singular solution concept to the higher order equation and three variables. The equation.1774 year, Lagrange has made a systematic study of the relation between singular solution and general solution. He gives the general method and the singular solution is the geometric interpretation of the envelope of the integral curve family. The second stage is the stage of the proper qualitative theory based on the definite solution. This period is a transition period in the history of mathematical development and the basis of mathematical analysis. The foundation, the concept of the group and the creation of the complex function are all at this time. The ordinary differential equation is deeply influenced by these new concepts and new methods. The main results of this stage are: in 1820s, Cauchy established the existence and uniqueness theorem of the solution of Cauchy's problem.1873, and Lipschitz put forward the famous "Li". Psz condition ", the existence of Cauchy's existence and uniqueness theorem is improved.1875 year and 1876 Cauchy, Lipschitz, piasia and Bi card successive approximation of ordinary differential equation, and so on. The third stage is the analytical theory stage of the development of ordinary differential equations. One of the main results of this stage is the use of power series and generalized power series. The solution of some important two order linear equations is obtained, and some very important special functions are obtained. The Riemann-Fuchs singularity theory is also a very important achievement in this stage. The fourth stage is the qualitative theory stage of the ordinary differential equation, and Poincare and Lyapunov create the qualitative theory and the differential equation of the differential equation respectively. The polynomial differential system is a simple and important ordinary differential equation. The study of the limit cycle problem occupies an important position in the qualitative theory of differential equations. In 1900, the second half of the sixteenth problems proposed by Hilbert is to discuss the maximum number and phase of the limit cycle of the plane polynomial system. The homogeneous polynomial differential system is an important class in the polynomial system. So far, the homogeneous polynomial differential system has many achievements. Markus has studied the P, the two homogeneous polynomial vector field (P, Q) of the Q mutual quality (P, Q) obtained the standard form of homogeneous multinomial differential system and the effective invariant of the system. The theory.Cima obtains the classification theorem and the classification of the algebraic characteristics of the four order two elements in the real number domain. The quasi homogeneous polynomial differential system is the generalization of the homogeneous polynomial differential system. In recent years, the quasi homogeneous system has attracted the attention of many scholars, such as the quasi homogeneous decomposition, the integrability of the homogeneous polynomial system, the central problem, the limit ring, the standard. The quasicrystals have all obtained rich results in.2013 years. Garcia gives an algorithm for the quasi homogeneous classification of planar quasi homogeneous polynomial systems, and uses the algorithm to get all the quasi homogeneous classification of the plane 2 and 3 polynomial systems. In this paper, the f (Q1, Q2) and G (Q1, Q2) in the Newtonian equations of motion are Q1, Q2 N and m, respectively. We first discuss some quasi homogeneous properties of the equation group (1) and then give the quasi homogeneous classification algorithm of (1). Finally, we use the algorithm to give the quasi homogeneous classification of M < n = 4 (1), remember the weight vector omega = (S1, S2, S3, S4, D), and the quasi homogeneous vector field (P1, P2, F, g). The following classification results are listed as follows:

【学位授予单位】：吉林大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

【参考文献】