更改

虽然这是大多数历史学家的观点，但少数人更进一步，比如[[E.T.Whittaker]]，他认为庞加莱和洛伦兹才是相对论的真正发现者<ref>Whittaker 1953, Secondary sources on relativity</ref>。

虽然这是大多数历史学家的观点，但少数人更进一步，比如[[E.T.Whittaker]]，他认为庞加莱和洛伦兹才是相对论的真正发现者<ref>Whittaker 1953, Secondary sources on relativity</ref>。

===Algebra and number theory===

===Algebra and number theory代数与数论===

After defending his doctoral thesis on the study of singular points of the system of differential equations, Poincaré wrote a series of memoirs under the title "On curves defined by differential equations" (1881–1882). In these articles, he built a new branch of mathematics, called "qualitative theory of differential equations". Poincaré showed that even if the differential equation can not be solved in terms of known functions, yet from the very form of the equation, a wealth of information about the properties and behavior of the solutions can be found. In particular, Poincaré investigated the nature of the trajectories of the integral curves in the plane, gave a classification of singular points (saddle, focus, center, node), introduced the concept of a limit cycle and the loop index, and showed that the number of limit cycles is always finite, except for some special cases. Poincaré also developed a general theory of integral invariants and solutions of the variational equations. For the finite-difference equations, he created a new direction – the asymptotic analysis of the solutions. He applied all these achievements to study practical problems of mathematical physics and celestial mechanics, and the methods used were the basis of its topological works.

After defending his doctoral thesis on the study of singular points of the system of differential equations, Poincaré wrote a series of memoirs under the title "On curves defined by differential equations" (1881–1882). In these articles, he built a new branch of mathematics, called "qualitative theory of differential equations". Poincaré showed that even if the differential equation can not be solved in terms of known functions, yet from the very form of the equation, a wealth of information about the properties and behavior of the solutions can be found. In particular, Poincaré investigated the nature of the trajectories of the integral curves in the plane, gave a classification of singular points (saddle, focus, center, node), introduced the concept of a limit cycle and the loop index, and showed that the number of limit cycles is always finite, except for some special cases. Poincaré also developed a general theory of integral invariants and solutions of the variational equations. For the finite-difference equations, he created a new direction – the asymptotic analysis of the solutions. He applied all these achievements to study practical problems of mathematical physics and celestial mechanics, and the methods used were the basis of its topological works.

在为自己关于微分方程系统的奇点研究的博士论文进行辩护之后，庞加莱写了一系列回忆录，题目是《关于微分方程定义的曲线》(1881-1882)。在这些文章中，他建立了一个新的数学分支，称为“定性微分方程理论”。表明，即使微分方程不能用已知函数来求解，但是从方程的形式，可以找到关于解的性质和行为的丰富信息。特别地，庞加莱研究了平面上积分曲线轨迹的性质，给出了奇点(鞍点、焦点、中心点、节点)的分类，引入了极限环和环指数的概念，并证明了除某些特殊情况外，极限环的个数总是有限的。庞加莱还提出了积分不变量和变分方程解的一般理论。对于有限差分方程，他创造了一个新的方向——解的渐近分析。他应用所有这些成就来研究数学物理和天体力学的实际问题，所使用的方法是其拓扑工作的基础。

在为自己关于微分方程系统的奇点研究的博士论文进行辩护之后，庞加莱写了一系列回忆录，题目是《关于微分方程定义的曲线》(1881-1882)。在这些文章中，他建立了一个新的数学分支，称为“定性微分方程理论”。表明，即使微分方程不能用已知函数来求解，但是从方程的形式，可以找到关于解的性质和行为的丰富信息。特别地，庞加莱研究了平面上积分曲线轨迹的性质，给出了<font color="#ff8000">奇点(鞍点、焦点、中心点、节点) Singular points (saddle, focus, center, node)</font>的分类，引入了<font color="#ff8000"> 极限环和环指数Limit cycle and  Loop index</font>的概念，并证明了除某些特殊情况外，<font color="#ff8000"> 极限环</font>的个数总是有限的。庞加莱还提出了<font color="#ff8000"> 积分不变量Integral invariants</font>和<font color="#ff8000"> 变分方程Variational equations</font>解的一般理论。对于<font color="#ff8000">有限差分方程 Finite-difference equations</font>，他创造了一个新的方向——解的<font color="#ff8000"> 渐近分析Asymptotic analysis</font>。他应用所有这些成就来研究数学物理和天体力学的实际问题，所使用的方法是其拓扑工作的基础。

Poincaré introduced [[group theory]] to physics, and was the first to study the group of [[Lorentz transformations]].<ref>Poincaré, Selected works in three volumes. page = 682{{full citation needed|date=September 2019}}</ref> He also made major contributions to the theory of discrete groups and their representations.

Poincaré introduced [[group theory]] to physics, and was the first to study the group of [[Lorentz transformations]].<ref>Poincaré, Selected works in three volumes. page = 682{{full citation needed|date=September 2019}}</ref> He also made major contributions to the theory of discrete groups and their representations.

[[Image:Mug and Torus morph.gif|right|frame |50px |<center>Topological transformation of the torus into a mug </center>]]

[[Image:Mug and Torus morph.gif|right|frame |50px |<center>Topological transformation of the torus into a mug </center>]]

<gallery caption="The singular points of the integral curves">

<gallery caption="The singular points of the integral curves">

积分曲线的奇点" >  

<gallery caption="积分曲线的奇点" >  

文件: Phase Portrait Sadle.svg | Saddle

文件: Phase Portrait Sadle.svg | Saddle

===Topology===

===Topology拓扑学===

  File: Phase Portrait Stable Focus.svg | Focus

  File: Phase Portrait Stable Focus.svg | Focus

The subject is clearly defined by [[Felix Klein]] in his "Erlangen Program" (1872): the geometry invariants of arbitrary continuous transformation, a kind of geometry. The term "topology" was introduced, as suggested by [[Johann Benedict Listing]], instead of previously used "Analysis situs". Some important concepts were introduced by [[Enrico Betti]] and [[Bernhard Riemann]]. But the foundation of this science, for a space of any dimension, was created by Poincaré. His first article on this topic appeared in 1894.{{sfn|Stillwell|2010|p=419-435}}

The subject is clearly defined by [[Felix Klein]] in his "Erlangen Program" (1872): the geometry invariants of arbitrary continuous transformation, a kind of geometry. The term "topology" was introduced, as suggested by [[Johann Benedict Listing]], instead of previously used "Analysis situs". Some important concepts were introduced by [[Enrico Betti]] and [[Bernhard Riemann]]. But the foundation of this science, for a space of any dimension, was created by Poincaré. His first article on this topic appeared in 1894.{{sfn|Stillwell|2010|p=419-435}}

  File: Phase portrait center.svg | Center

  File: Phase portrait center.svg | Center

His research in geometry led to the abstract topological definition of [[homotopy]] and [[Homology (mathematics)|homology]]. He also first introduced the basic concepts and invariants of combinatorial topology, such as Betti numbers and the [[fundamental group]]. Poincaré proved a formula relating the number of edges, vertices and faces of ''n''-dimensional polyhedron (the Euler–Poincaré theorem) and gave the first precise formulation of the intuitive notion of dimension.<ref>{{citation|last=Aleksandrov|first=Pavel S. |authorlink=Pavel Alexandrov|title= Poincaré and topology| pages = 27–81}}{{full citation needed|date=September 2019}}</ref>

His research in geometry led to the abstract topological definition of [[homotopy]] and [[Homology (mathematics)|homology]]. He also first introduced the basic concepts and invariants of combinatorial topology, such as Betti numbers and the [[fundamental group]]. Poincaré proved a formula relating the number of edges, vertices and faces of ''n''-dimensional polyhedron (the Euler–Poincaré theorem) and gave the first precise formulation of the intuitive notion of dimension.<ref>{{citation|last=Aleksandrov|first=Pavel S. |authorlink=Pavel Alexandrov|title= Poincaré and topology| pages = 27–81}}{{full citation needed|date=September 2019}}</ref>

</gallery>

</gallery>

</gallery >  

</gallery >

 

 

===Astronomy and celestial mechanics===

===Astronomy and celestial mechanics===