本文選題：牛頓運(yùn)動(dòng)方程組 + 多項(xiàng)式微分系統(tǒng)�。� 參考：《吉林大學(xué)》2017年碩士論文

【摘要】：常微分方程是伴隨著微積分發(fā)展起來(lái)的,其成長(zhǎng)于生產(chǎn)實(shí)踐和數(shù)學(xué)的發(fā)展進(jìn)程,蘊(yùn)含著豐富的數(shù)學(xué)思想方法.它在天體力學(xué)和其它力學(xué)領(lǐng)域顯示出巨大的功能.牛頓通過(guò)解微分方程證實(shí)了地球繞太陽(yáng)的運(yùn)動(dòng)軌道是一個(gè)橢圓;海王星的存在是天文學(xué)家先通過(guò)微分方程的方法推算出來(lái),然后才實(shí)際觀測(cè)到的.常微分方程的形成和發(fā)展與力學(xué)、天文學(xué)、物理學(xué)以及其他科學(xué)技術(shù)的發(fā)展也有著密切的聯(lián)系.數(shù)學(xué)的其他分支的新發(fā)展,如復(fù)變函數(shù)、李群、組合拓?fù)鋵W(xué)等,都對(duì)常微分方程的發(fā)展產(chǎn)生了深刻的影響.目前,常微分方程在所有自然科學(xué)領(lǐng)域和眾多社會(huì)科學(xué)領(lǐng)域都有著廣泛的應(yīng)用.可以預(yù)測(cè)隨著社會(huì)技術(shù)的發(fā)展和需求,常微分方程會(huì)有更大的發(fā)展.常微分方程的發(fā)展經(jīng)歷了四個(gè)階段:第一階段是以求通解為主要內(nèi)容的經(jīng)典理論階段.1690年,Bernoulli James研究了與鐘擺運(yùn)動(dòng)有關(guān)的“等時(shí)曲線問(wèn)題(在相等的時(shí)間內(nèi),使擺沿著這條曲線作一次完全的振動(dòng)(不考慮擺所經(jīng)歷的弧長(zhǎng)的大小))”.他通過(guò)分析建立了常微分方程的模型,并用分離變量法解出了這條擺線的方程.1690年,Bernoul-1i James 提出了“懸鏈線問(wèn)題(繩子懸掛于兩固定點(diǎn)而形成的曲線(繩子是柔軟的但不能是伸長(zhǎng)的))”.Bernoulli John和Leibniz用微積分的方法解決了懸鏈線問(wèn)題.后來(lái)又研究了等角軌線問(wèn)題,正交軌線問(wèn)題等等.1691年,Leibniz給出了變量分離法.1694年,他使用了常數(shù)變易法把一階常微分方程化成積分,又發(fā)現(xiàn)了方程的一個(gè)解族的包絡(luò)也是解.1695年,Bernoulli John給出著名的Bernoulli方程.Leibniz用變換將其化為線性方程.1715-1718年,Taylor討論微分方程的奇解、包絡(luò)和變量代換公式.1734年,Clairaut研究了 Clairaut方程,發(fā)現(xiàn)這個(gè)方程的通解是直線族,而直線的包絡(luò)線就是奇解,Clairaut和Euler對(duì)奇解進(jìn)行了全面的研究,給出從微分方程本身求奇解的方法.1734年,Euler給出了恰當(dāng)方程的定義.他與克萊羅各自找到了方程是恰當(dāng)方程的條件,并發(fā)現(xiàn)若方程是恰當(dāng)?shù)?則方程是可積的.1739年克萊羅提出了積分因子的概念,Euler確定了可采用積分因子的的方法求解方程.1772年,Laplace將奇解概念推廣到高階方程和三個(gè)變量的方程.1774年,Lagrange對(duì)奇解和通解的聯(lián)系作了系統(tǒng)的研究,他給出了一般的方法及奇解是積分曲線族包絡(luò)的幾何解釋等等.第二階段是以定解問(wèn)題為研究?jī)?nèi)容的適定性理論階段.此時(shí)期是數(shù)學(xué)發(fā)展史上的一個(gè)轉(zhuǎn)變時(shí)期,數(shù)學(xué)分析的基礎(chǔ)、群的概念、復(fù)變函數(shù)的開(kāi)創(chuàng)等都在這個(gè)時(shí)期,常微分方程深受這些新概念和新方法的影響,進(jìn)入了它發(fā)展的第二個(gè)階段.這一階段的主要結(jié)果有:19世紀(jì)20年代,柯西建立了柯西問(wèn)題解的存在唯一性定理.1873年,李普希茲提出著名的“李普希茲條件”,對(duì)柯西的存在唯一性定理作了改進(jìn).1875年和1876年柯西、李普希茲、皮亞拿和比卡先后給出常微分方程的逐次逼近法等等.第三階段是常微分方程發(fā)展的解析理論階段.這一階段的主要結(jié)果之一是運(yùn)用冪級(jí)數(shù)和廣義冪級(jí)數(shù)解法,求出一些重要的二階線性方程的級(jí)數(shù)解,并得到極其重要的一些特殊函數(shù),Riemann-Fuchs奇點(diǎn)理論也是這一階段非常重要的成果.第四階段是常微分方程的定性理論階段,龐加萊和李雅普諾夫分別開(kāi)創(chuàng)了微分方程定性理論和微分方程運(yùn)動(dòng)穩(wěn)定性理論.多項(xiàng)式微分系統(tǒng)是一類簡(jiǎn)單而又重要的常微分方程,極限環(huán)問(wèn)題的研究在微分方程的定性理論中占有很重要的地位,1900年,Hilbert提出的第十六個(gè)問(wèn)題的后半部分就是討論平面多項(xiàng)式系統(tǒng)的極限環(huán)的最多個(gè)數(shù)和相對(duì)位置.齊次多項(xiàng)式微分系統(tǒng)作為多項(xiàng)式系統(tǒng)中重要的一類.到目前為止,齊次多項(xiàng)式微分系統(tǒng)已有不少的成果,Markus研究了P,Q互質(zhì)的二次齊次多項(xiàng)式向量場(chǎng)(P,Q)的分類.Algaba得到了齊次多項(xiàng)式微分系統(tǒng)的標(biāo)準(zhǔn)型以及系統(tǒng)有效的不變量理論.Cima得到實(shí)數(shù)域上四階二元型的分類定理和代數(shù)特征的分類形式.擬齊次多項(xiàng)式微分系統(tǒng)是齊次多項(xiàng)式微分系統(tǒng)的推廣.近年來(lái),擬齊次系統(tǒng)受到眾多學(xué)者的關(guān)注,例如:擬齊次分解,擬齊次多項(xiàng)式系統(tǒng)的可積性,中心問(wèn)題,極限環(huán),標(biāo)準(zhǔn)型等都取得了豐富的成果.2013年,Garcia給出了一種對(duì)平面擬齊次多項(xiàng)式系統(tǒng)進(jìn)行擬齊次分類的算法,并利用該算法得到了平面2次和3次多項(xiàng)式系統(tǒng)的所有擬齊次分類.本文考慮牛頓運(yùn)動(dòng)方程組其中f(q1,q2)和g(q1,q2)分別是q1,q2的n階和m階多項(xiàng)式.我們首先探討方程組(1)的一些擬齊次性質(zhì),然后給出(1)的擬齊次分類算法,最后利用算法給出當(dāng)m ≤ n = 4時(shí)(1)的擬齊次分類,記權(quán)向量ω =(s1,s2,s3,s4,d),擬齊次向量場(chǎng)為(P1,p2,f,g).分類結(jié)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:

【學(xué)位授予單位】：吉林大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175

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