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===Astronomy and celestial mechanics===
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===Astronomy and celestial mechanics天文学与天体力学===
    
[[File:N-body problem (3).gif|frame|left|150px | <center> Chaotic motion in three-body problem (computer simulation).</center>]]
 
[[File:N-body problem (3).gif|frame|left|150px | <center> Chaotic motion in three-body problem (computer simulation).</center>]]
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Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). They introduced the small parameter method, fixed points, integral invariants, variational equations, the convergence of the asymptotic expansions. Generalizing a theory of Bruns (1887), Poincaré showed that the three-body problem is not integrable. In other words, the general solution of the three-body problem can not be expressed in terms of algebraic and transcendental functions through unambiguous coordinates and velocities of the bodies. His work in this area was the first major achievement in celestial mechanics since [[Isaac Newton]].<ref>J. Stillwell, Mathematics and its history, [https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA254 page 254]</ref>
 
Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). They introduced the small parameter method, fixed points, integral invariants, variational equations, the convergence of the asymptotic expansions. Generalizing a theory of Bruns (1887), Poincaré showed that the three-body problem is not integrable. In other words, the general solution of the three-body problem can not be expressed in terms of algebraic and transcendental functions through unambiguous coordinates and velocities of the bodies. His work in this area was the first major achievement in celestial mechanics since [[Isaac Newton]].<ref>J. Stillwell, Mathematics and its history, [https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA254 page 254]</ref>
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庞加莱出版了两本经典专著《天体力学的新方法》(1892-1899)和《天体力学讲座》(1905-1910)。其中,他成功地将他们的研究成果应用于三体的运动问题,并详细研究了解的行为(频率、稳定性、渐近性等)。介绍了<font color="#ff8000">小参数法、不动点、积分不变量、变分方程、渐近展开的收敛性</font>。推广Bruns(1887)的一个理论,庞加莱证明了<font color="#ff8000"> 三体问题</font>是不可积的。换言之,<font color="#ff8000"> 三体问题</font>的一般解不能通过物体的明确坐标和速度用代数函数和超越函数来表示。他在这方面的工作是自[[艾萨克牛顿]]以来在天体力学方面的第一个重大成就。<ref>J. Stillwell, Mathematics and its history, [https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA254 page 254]</ref>
    
Poincaré's work habits have been compared to a bee flying from flower to flower. Poincaré was interested in the way his mind worked; he studied his habits and gave a talk about his observations in 1908 at the Institute of General Psychology in Paris. He linked his way of thinking to how he made several discoveries.
 
Poincaré's work habits have been compared to a bee flying from flower to flower. Poincaré was interested in the way his mind worked; he studied his habits and gave a talk about his observations in 1908 at the Institute of General Psychology in Paris. He linked his way of thinking to how he made several discoveries.
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These monographs include an idea of Poincaré, which later became the basis for mathematical "[[chaos theory]]" (see, in particular, the [[Poincaré recurrence theorem]]) and the general theory of [[dynamical system]]s.
 
These monographs include an idea of Poincaré, which later became the basis for mathematical "[[chaos theory]]" (see, in particular, the [[Poincaré recurrence theorem]]) and the general theory of [[dynamical system]]s.
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这些专著包括了庞加莱的思想,这后来成为数学“[[混沌理论]]”的基础(特别参见[[庞加莱递推定理]])和[[动力系统]]s的一般理论。
    
The mathematician Darboux claimed he was un intuitif (an intuitive), arguing that this is demonstrated by the fact that he worked so often by visual representation. He did not care about being rigorous and disliked logic. (Despite this opinion, Jacques Hadamard wrote that Poincaré's research demonstrated marvelous clarity and Poincaré himself wrote that he <!-- TODO: Add Poincaré's opinion on rigorousness, see http://www.forgottenbooks.org/readbook/American_Journal_of_Mathematics_1890_v12_1000084889#233 — Each time I can I'm absolute rigour --> believed that logic was not a way to invent but a way to structure ideas and that logic limits ideas.)
 
The mathematician Darboux claimed he was un intuitif (an intuitive), arguing that this is demonstrated by the fact that he worked so often by visual representation. He did not care about being rigorous and disliked logic. (Despite this opinion, Jacques Hadamard wrote that Poincaré's research demonstrated marvelous clarity and Poincaré himself wrote that he <!-- TODO: Add Poincaré's opinion on rigorousness, see http://www.forgottenbooks.org/readbook/American_Journal_of_Mathematics_1890_v12_1000084889#233 — Each time I can I'm absolute rigour --> believed that logic was not a way to invent but a way to structure ideas and that logic limits ideas.)
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Poincaré authored important works on astronomy for the equilibrium figures of a gravitating rotating fluid. He introduced the important concept of bifurcation points and proved the existence of equilibrium figures such as the non-ellipsoids, including ring-shaped and pear-shaped figures, and their stability. For this discovery, Poincaré received the Gold Medal of the Royal Astronomical Society (1900).<ref>A. Kozenko, The theory of planetary figures, pages = 25–26{{full citation needed|date=September 2019}}</ref>
 
Poincaré authored important works on astronomy for the equilibrium figures of a gravitating rotating fluid. He introduced the important concept of bifurcation points and proved the existence of equilibrium figures such as the non-ellipsoids, including ring-shaped and pear-shaped figures, and their stability. For this discovery, Poincaré received the Gold Medal of the Royal Astronomical Society (1900).<ref>A. Kozenko, The theory of planetary figures, pages = 25–26{{full citation needed|date=September 2019}}</ref>
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庞加莱为引力旋转流体的平衡图写了重要的天文学著作。他引入了分支点的重要概念,证明了非椭球体(包括环形和梨形)等平衡图形的存在性及其稳定性。这枚天文发现奖(1900年)被英国皇家天文学会授予。<ref>A. Kozenko, The theory of planetary figures, pages = 25–26{{full citation needed|date=September 2019}}</ref>
    
===Differential equations and mathematical physics===
 
===Differential equations and mathematical physics===
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