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亨利·庞加莱 Jules Henri Poincaré (查看源代码)

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{{short description|French mathematician, physicist, engineer, and philosopher of science}}

{{More citations needed|date=April 2017}}

{{Use dmy dates|date=November 2020}}



{{Infobox scientist

{{Infobox scientist

{信息盒科学家

|name = Henri Poincaré

|name = Henri Poincaré

|name = Henri Poincaré

|other_names = Jules Henri Poincaré

|other_names = Jules Henri Poincaré

其他名字 = 儒勒·昂利·庞加莱

|image = PSM V82 D416 Henri Poincare.png

|image = PSM V82 D416 Henri Poincare.png

82 D416 Henri Poincare.png

|caption = Henri Poincaré <br />(photograph published in 1913)

|caption = Henri Poincaré <br />(photograph published in 1913)

摄于1913年

|birth_date = {{birth date|df=yes|1854|4|29}}

|birth_date =

出生日期

|birth_place = [[Nancy, France|Nancy]], [[Meurthe-et-Moselle]], France

|birth_place = Nancy, Meurthe-et-Moselle, France

出生地: 南希，默尔特-摩泽尔省，法国

|death_date = {{death date and age|df=yes|1912|7|17|1854|4|29}}

|death_date =

死亡日期

|death_place = [[Paris]], France

|death_place = Paris, France

死亡地点: 法国巴黎

|residence = France

|residence = France

居住地: 法国

|nationality = [[French people|French]]

|nationality = French

| 国籍: 法国

|fields = Mathematics and [[physics]]

|fields = Mathematics and physics

| fields = 数学和物理

|workplaces = {{plainlist|

|workplaces = {{plainlist|

工作场所 = { plainlist |

*[[Corps des Mines]]

*[[Caen University]]

*[[University of Paris|La Sorbonne]]

*[[Bureau des Longitudes]]}}

|education = {{plainlist|

|education = {{plainlist|

2009年10月11日

*Lycée Nancy (now {{ill|Lycée Henri-Poincaré|fr}})

*[[École Polytechnique]]

*[[École des Mines]]

*[[University of Paris]] ([[Doctorat|Dr]], 1879)}}

|thesis_title = Sur les propriétés des fonctions définies par les équations différences

|thesis_title = Sur les propriétés des fonctions définies par les équations différences

|thesis_title = Sur les propriétés des fonctions définies par les équations différences

|thesis_url = https://web.archive.org/web/20160506152142/https://iris.univ-lille1.fr/handle/1908/458

|thesis_url = https://web.archive.org/web/20160506152142/https://iris.univ-lille1.fr/handle/1908/458

Https://web.archive.org/web/20160506152142/https://iris.univ-lille1.fr/handle/1908/458

|thesis_year = 1879

|thesis_year = 1879

论文年份 = 1879

|doctoral_advisor = [[Charles Hermite]]

|doctoral_advisor = Charles Hermite

博士生导师查尔斯 · 赫米特

|academic_advisors =

|academic_advisors =

学术顾问

|doctoral_students = {{plainlist|

|doctoral_students = {{plainlist|

博士生 = { plainlist |

*[[Louis Bachelier]]

*[[Jean Bosler]]

*[[Dimitrie Pompeiu]]

*[[Mihailo Petrović]]}}

|notable_students = {{plainlist|

|notable_students = {{plainlist|

2012年10月12日

*[[Tobias Dantzig]]

*[[Théophile de Donder]]}}

|known_for = {{plainlist|

|known_for = {{plainlist|

2009年10月11日

*[[Poincaré conjecture]]

*[[Three-body problem]]

*[[Topology]]

*[[Special relativity]]

*[[Poincaré–Hopf theorem]]

*[[Poincaré duality]]

*{{nowrap|[[Poincaré–Birkhoff–Witt theorem]]}}

*[[Poincaré inequality]]

*[[Hilbert–Poincaré series]]

*[[Poincaré series (modular form)|Poincaré series]]

*[[Poincaré metric]]

*[[Rotation number]]

*[[Fundamental group]]

*[[Betti number|Coining the term "Betti number"]]

*[[Bifurcation theory]]

*[[Chaos theory]]

*[[Brouwer fixed-point theorem]]

*[[Sphere-world]]

*[[Poincaré–Bendixson theorem]]

*[[Poincaré–Lindstedt method]]

*[[Poincaré recurrence theorem]]

*[[Kelvin's circulation theorem#Poincaré–Bjerknes circulation theorem|Poincaré–Bjerknes circulation theorem]]

*[[Poincaré group]]

*[[Poincaré gauge]]

*[[French historical epistemology]]

*[[Preintuitionism]]/[[conventionalism]]

*[[Predicativism]]

}}

}}

}}

|influences = {{plainlist|

|influences = {{plainlist|

2009年10月11日

*[[Lazarus Fuchs]]

*[[Immanuel Kant]]<ref>[http://www.iep.utm.edu/poi-math/#H3 "Poincaré's Philosophy of Mathematics"], entry in the [[Internet Encyclopedia of Philosophy]].</ref>

*[[Ernst Mach]]<ref>[https://plato.stanford.edu/entries/poincare/ "Henri Poincaré"], entry in the [[Stanford Encyclopedia of Philosophy]].</ref>}}

|influenced = {{plainlist|

|influenced = {{plainlist|

2009年10月11日

*[[Louis Rougier]]

*[[George David Birkhoff]]

[[Albert Einstein]]<ref>Einstein's letter to Michele Besso, Princeton, 6 March 1952</ref>}}

Albert Einstein}}

阿尔伯特 · 爱因斯坦

|awards = {{plainlist|

|awards = {{plainlist|

2012年10月12日

*{{nowrap|[[Gold Medal of the Royal Astronomical Society|RAS Gold Medal]] (1900)}}

*[[Sylvester Medal]] (1901)

*[[Matteucci Medal]] (1905)

*[[Bolyai Prize]] (1905)

*[[Bruce Medal]] (1911)}}

|signature = Henri Poincaré Signature.svg

|signature = Henri Poincaré Signature.svg

签名: Henri poincaré Signature.svg

|footnotes = He was an uncle of [[Pierre Boutroux]].

|footnotes = He was an uncle of Pierre Boutroux.

他是皮埃尔 · 布特鲁克斯的叔叔。

}}

}}

}}



'''Jules Henri Poincaré''' ({{IPAc-en|UK|ˈ|p|w|æ̃|k|ɑr|eɪ}}<ref>{{OED|Poincaré}}</ref> [US: stress final syllable], {{IPA-fr|ɑ̃ʁi pwɛ̃kaʁe|lang|Fr-Henri Poincaré.ogg}};<ref name="forvo">{{cite web|url=http://www.forvo.com/word/poincar%C3%A9/ |title=Poincaré pronunciation: How to pronounce Poincaré in French |website=forvo.com|accessdate=}}</ref><ref name="pronouncekiwi">{{cite web|url=http://www.pronouncekiwi.com/Henri%20Poincaré |title=How To Pronounce Henri Poincaré |website=pronouncekiwi.com|accessdate=}}</ref> 29 April 1854&nbsp;– 17 July 1912) was a French [[mathematician]], [[theoretical physicist]], [[engineer]], and [[philosophy of science|philosopher of science]]. He is often described as a [[polymath]], and in mathematics as "The Last Universalist",<ref>{{cite book | first1=J. M. | last1=Ginoux | first2=C. | last2=Gerini | title=Henri Poincaré: A Biography Through the Daily Papers | publisher=World Scientific | date=2013 | isbn=978-981-4556-61-3 | doi=10.1142/8956 }}</ref> since he excelled in all fields of the discipline as it existed during his lifetime.

Jules Henri Poincaré ( [US: stress final syllable], ; 29 April 1854&nbsp;– 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The Last Universalist", since he excelled in all fields of the discipline as it existed during his lifetime.

儒勒·昂利·庞加莱是法国数学家、理论物理学家、工程师和科学哲学家。他经常被描述为一个博学者，在数学方面被称为“最后的普遍主义者” ，因为他在所有领域的学科，因为它存在于他的一生。



As a mathematician and physicist, he made many original fundamental contributions to [[Pure mathematics|pure]] and [[applied mathematics]], [[mathematical physics]], and [[celestial mechanics]].<ref>{{cite journal|author=Hadamard, Jacques|authorlink=Jacques Hadamard|title=The early scientific work of Henri Poincaré|journal=The Rice Institute Pamphlet|date=July 1922|volume=9|issue=3|pages=111–183|url=http://catalog.hathitrust.org/Record/100592035}}</ref> In his research on the [[three-body problem]], Poincaré became the first person to discover a chaotic [[deterministic system]] which laid the foundations of modern [[chaos theory]]. He is also considered to be one of the founders of the field of [[topology]].

As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is also considered to be one of the founders of the field of topology.

作为一名数学家和物理学家，他对纯粹数学和应用数学、数学物理学和天体力学做出了许多原创性的基础性贡献。在他对三体的研究中，庞加莱成为第一个发现混沌确定性模型的人，它奠定了现代混沌理论的基础。他也被认为是拓扑学领域的创始人之一。



Poincaré made clear the importance of paying attention to the invariance of laws of physics under different transformations, and was the first to present the [[Lorentz transformations]] in their modern symmetrical form. Poincaré discovered the remaining relativistic velocity transformations and recorded them in a letter to [[Hendrik Lorentz]] in 1905. Thus he obtained perfect invariance of all of [[Maxwell's equations]], an important step in the formulation of the theory of [[special relativity]]. In 1905, Poincaré first proposed [[gravitational wave]]s (''ondes gravifiques'') emanating from a body and propagating at the speed of light as being required by the Lorentz transformations.

Poincaré made clear the importance of paying attention to the invariance of laws of physics under different transformations, and was the first to present the Lorentz transformations in their modern symmetrical form. Poincaré discovered the remaining relativistic velocity transformations and recorded them in a letter to Hendrik Lorentz in 1905. Thus he obtained perfect invariance of all of Maxwell's equations, an important step in the formulation of the theory of special relativity. In 1905, Poincaré first proposed gravitational waves (ondes gravifiques) emanating from a body and propagating at the speed of light as being required by the Lorentz transformations.

庞加莱阐明了物理定律在不同变换下的不变性的重要性，并率先提出了洛伦兹变换的现代对称形式。庞加莱发现了剩下的相对论速度变换，并在1905年写给亨德里克 · 洛伦兹的信中记录了它们。因此，他得到了所有麦克斯韦方程的完美不变性，这是狭义相对论理论形成过程中的重要一步。1905年，庞加莱首次提出引力波(ondes 引力波) ，它从物体中发射出来，并按照洛伦兹变换的要求以光速传播。



The [[Poincaré group]] used in physics and mathematics was named after him.

The Poincaré group used in physics and mathematics was named after him.

用于物理和数学的庞加莱小组就是以他的名字命名的。



Early in the 20th century he formulated the [[Poincaré conjecture]] that became over time one of the famous [[unsolved problems in mathematics]] until it was solved in 2002–2003 by [[Grigori Perelman]].

Early in the 20th century he formulated the Poincaré conjecture that became over time one of the famous unsolved problems in mathematics until it was solved in 2002–2003 by Grigori Perelman.

在20世纪早期，他制定了庞加莱猜想，随着时间的推移，这成为著名的未解决的数学问题之一，直到2002年至2003年被格里戈里·佩雷尔曼解决。



==Life==

Poincaré was born on 29 April 1854 in Cité Ducale neighborhood, [[Nancy, Meurthe-et-Moselle]], into an influential French family.<ref>Belliver, 1956</ref> His father Léon Poincaré (1828–1892) was a professor of medicine at the [[University of Nancy]].<ref>Sagaret, 1911</ref> His younger sister Aline married the spiritual philosopher [[Émile Boutroux]]. Another notable member of Henri's family was his cousin, [[Raymond Poincaré]], a fellow member of the [[Académie française]], who would serve as President of France from 1913 to 1920.<ref name="IEP">[http://www.utm.edu/research/iep/p/poincare.htm The Internet Encyclopedia of Philosophy] Jules Henri Poincaré article by Mauro Murzi&nbsp;– Retrieved November 2006.</ref>

Poincaré was born on 29 April 1854 in Cité Ducale neighborhood, Nancy, Meurthe-et-Moselle, into an influential French family. His father Léon Poincaré (1828–1892) was a professor of medicine at the University of Nancy. His younger sister Aline married the spiritual philosopher Émile Boutroux. Another notable member of Henri's family was his cousin, Raymond Poincaré, a fellow member of the Académie française, who would serve as President of France from 1913 to 1920.

1854年4月29日，庞加莱出生在 Cité Ducale 一个有影响力的法国家庭，家住默尔特-摩泽尔省南希。他的父亲 Léon poincaré (1828-1892)是南希大学的医学教授。他的妹妹艾琳嫁给了精神哲学家埃米尔 · 布特鲁克斯。的家族中另一个著名的成员是他的表弟，雷蒙·普恩加莱，法兰西学术院的同事，他在1913年到1920年间担任法国总统。



===Education===

[[File:Henri Poincaré maison natale Nancy plaque.jpg|thumb|right|200px| Plaque on the birthplace of Henri Poincaré at house number 117 on the Grande Rue in the city of Nancy]]

Plaque on the birthplace of Henri Poincaré at house number 117 on the Grande Rue in the city of Nancy

位于南希市大街117号的昂利 · 庞加莱出生地牌匾



During his childhood he was seriously ill for a time with [[diphtheria]] and received special instruction from his mother, Eugénie Launois (1830–1897).

During his childhood he was seriously ill for a time with diphtheria and received special instruction from his mother, Eugénie Launois (1830–1897).

在童年时期，他曾一度患有严重的白喉病，并接受了他母亲欧热尼 · 劳诺伊斯(Eugénie Launois，1830-1897)的特别指导。



In 1862, Henri entered the Lycée in [[Nancy, Meurthe-et-Moselle|Nancy]] (now renamed the {{ill|Lycée Henri-Poincaré|fr}} in his honour, along with [[Henri Poincaré University]], also in Nancy). He spent eleven years at the Lycée and during this time he proved to be one of the top students in every topic he studied. He excelled in written composition. His mathematics teacher described him as a "monster of mathematics" and he won first prizes in the [[concours général]], a competition between the top pupils from all the Lycées across France. His poorest subjects were music and physical education, where he was described as "average at best".<ref>O'Connor et al., 2002</ref> However, poor eyesight and a tendency towards absentmindedness may explain these difficulties.<ref>Carl, 1968</ref> He graduated from the Lycée in 1871 with a bachelor's degree in letters and sciences.

In 1862, Henri entered the Lycée in Nancy (now renamed the in his honour, along with Henri Poincaré University, also in Nancy). He spent eleven years at the Lycée and during this time he proved to be one of the top students in every topic he studied. He excelled in written composition. His mathematics teacher described him as a "monster of mathematics" and he won first prizes in the concours général, a competition between the top pupils from all the Lycées across France. His poorest subjects were music and physical education, where he was described as "average at best". However, poor eyesight and a tendency towards absentmindedness may explain these difficulties. He graduated from the Lycée in 1871 with a bachelor's degree in letters and sciences.

1862年，亨利进入南希的 Lycée (为了纪念他，现在与同样位于南希的亨利 · 庞加莱大学一起重新命名为南希大学)。他在 Lycée 学习了11年，在此期间，他证明自己在所学的每一个领域都是最优秀的学生之一。他的作文写得很好。他的数学老师形容他是一个“数学怪兽” ，他在总决赛中获得了一等奖，总决赛是法国所有中学的优秀学生之间的比赛。他最差的科目是音乐和体育，在那里他被描述为“最好的平均水平”。然而，视力差和心不在焉的倾向也许可以解释这些困难。他于1871年毕业于 Lycée，获得文理学学士学位。



During the [[Franco-Prussian War]] of 1870, he served alongside his father in the Ambulance Corps.

During the Franco-Prussian War of 1870, he served alongside his father in the Ambulance Corps.

1870年的普法战争，他和父亲一起在救护队服役。



Poincaré entered the [[École Polytechnique]] as the top qualifier in 1873 and graduated in 1875. There he studied mathematics as a student of [[Charles Hermite]], continuing to excel and publishing his first paper (''Démonstration nouvelle des propriétés de l'indicatrice d'une surface'') in 1874. From November 1875 to June 1878 he studied at the [[École des Mines]], while continuing the study of mathematics in addition to the mining engineering syllabus, and received the degree of ordinary mining engineer in March 1879.<ref>F. Verhulst</ref>

Poincaré entered the École Polytechnique as the top qualifier in 1873 and graduated in 1875. There he studied mathematics as a student of Charles Hermite, continuing to excel and publishing his first paper (Démonstration nouvelle des propriétés de l'indicatrice d'une surface) in 1874. From November 1875 to June 1878 he studied at the École des Mines, while continuing the study of mathematics in addition to the mining engineering syllabus, and received the degree of ordinary mining engineer in March 1879.

1873年，庞加莱以最高资格进入巴黎综合理工学院，1875年毕业。在那里，他作为查尔斯 · 赫米特的学生学习了数学，继续超越并在1874年发表了他的第一篇论文(新标题《表面的所有权》)。从1875年11月至1878年6月，他在矿业学院学习，同时继续学习采矿工程教学大纲以外的数学，并于1879年3月获得普通采矿工程师学位。



As a graduate of the École des Mines, he joined the [[Corps des Mines]] as an inspector for the [[Vesoul]] region in northeast France. He was on the scene of a mining disaster at [[Magny-lès-Jussey|Magny]] in August 1879 in which 18 miners died. He carried out the official investigation into the accident in a characteristically thorough and humane way.

As a graduate of the École des Mines, he joined the Corps des Mines as an inspector for the Vesoul region in northeast France. He was on the scene of a mining disaster at Magny in August 1879 in which 18 miners died. He carried out the official investigation into the accident in a characteristically thorough and humane way.

作为矿业学院的毕业生，他加入了矿业部队，担任法国东北部沃苏勒地区的监察员。1879年8月，他在马尼矿难现场，18名矿工遇难。他以典型的彻底和人道的方式对事故进行了官方调查。



At the same time, Poincaré was preparing for his [[Doctorate in Science]] in mathematics under the supervision of Charles Hermite. His doctoral thesis was in the field of [[differential equations]]. It was named ''Sur les propriétés des fonctions définies par les équations aux différences partielles''. Poincaré devised a new way of studying the properties of these equations. He not only faced the question of determining the integral of such equations, but also was the first person to study their general geometric properties. He realised that they could be used to model the behaviour of multiple bodies in free motion within the [[solar system]]. Poincaré graduated from the [[University of Paris]] in 1879.

At the same time, Poincaré was preparing for his Doctorate in Science in mathematics under the supervision of Charles Hermite. His doctoral thesis was in the field of differential equations. It was named Sur les propriétés des fonctions définies par les équations aux différences partielles. Poincaré devised a new way of studying the properties of these equations. He not only faced the question of determining the integral of such equations, but also was the first person to study their general geometric properties. He realised that they could be used to model the behaviour of multiple bodies in free motion within the solar system. Poincaré graduated from the University of Paris in 1879.

与此同时，庞加莱在查尔斯 · 埃尔米特的指导下正在准备他的数学博士学位。他的博士论文是在微分方程领域。It was named Sur les propriétés des fonctions définies par les équations aux différences partielles.庞加莱设计了一种研究这些方程性质的新方法。他不仅面临着确定这些方程的积分的问题，而且是第一个研究它们的一般几何性质的人。他意识到，这些物质可以用来模拟太阳系内自由运动的多个物体的行为。庞加莱1879年毕业于巴黎大学。



[[Image:Young Poincare.jpg|left|upright|thumb|The young Henri Poincaré]]

The young Henri Poincaré

年轻的亨利 · 庞加莱



===First scientific achievements===

After receiving his degree, Poincaré began teaching as junior lecturer in mathematics at the [[Caen University|University of Caen]] in Normandy (in December 1879). At the same time he published his first major article concerning the treatment of a class of [[automorphic function]]s.

After receiving his degree, Poincaré began teaching as junior lecturer in mathematics at the University of Caen in Normandy (in December 1879). At the same time he published his first major article concerning the treatment of a class of automorphic functions.

获得学位后，庞加莱开始在诺曼底的卡昂大学担任数学初级讲师(1879年12月)。与此同时，他发表了第一篇关于一类自守函数的处理的重要文章。



There, in [[Caen]], he met his future wife, Louise Poulain d'Andecy and on 20 April 1881, they married. Together they had four children: Jeanne (born 1887), Yvonne (born 1889), Henriette (born 1891), and Léon (born 1893).

There, in Caen, he met his future wife, Louise Poulain d'Andecy and on 20 April 1881, they married. Together they had four children: Jeanne (born 1887), Yvonne (born 1889), Henriette (born 1891), and Léon (born 1893).

在那里，他在卡昂遇到了他未来的妻子路易丝 · 普兰 · 德安德西，并于1881年4月20日结婚。他们共有四个孩子: 珍妮(生于1887年)、伊冯娜(生于1889年)、亨利埃特(生于1891年)和莱昂(生于1893年)。



Poincaré immediately established himself among the greatest mathematicians of Europe, attracting the attention of many prominent mathematicians. In 1881 Poincaré was invited to take a teaching position at the Faculty of Sciences of the [[University of Paris]]; he accepted the invitation. During the years of 1883 to 1897, he taught mathematical analysis in [[École Polytechnique]].

Poincaré immediately established himself among the greatest mathematicians of Europe, attracting the attention of many prominent mathematicians. In 1881 Poincaré was invited to take a teaching position at the Faculty of Sciences of the University of Paris; he accepted the invitation. During the years of 1883 to 1897, he taught mathematical analysis in École Polytechnique.

庞加莱立即成为欧洲最伟大的数学家之一，吸引了许多杰出数学家的注意。1881年，庞加莱应邀到巴黎大学科学院任教，他接受了邀请。在1883年到1897年间，他在巴黎综合理工学院教授数学分析。



In 1881–1882, Poincaré created a new branch of mathematics: [[qualitative theory of differential equations]]. He showed how it is possible to derive the most important information about the behavior of a family of solutions without having to solve the equation (since this may not always be possible). He successfully used this approach to problems in [[celestial mechanics]] and [[mathematical physics]].

In 1881–1882, Poincaré created a new branch of mathematics: qualitative theory of differential equations. He showed how it is possible to derive the most important information about the behavior of a family of solutions without having to solve the equation (since this may not always be possible). He successfully used this approach to problems in celestial mechanics and mathematical physics.

1881-1882年，庞加莱创立了一个新的数学分支: 微分方程定性理论。他展示了如何不用解方程就可以得到关于一组解的行为的最重要的信息(因为这可能并不总是可能的)。他成功地用这种方法解决了天体力学和数学物理的问题。



===Career===

He never fully abandoned his mining career to mathematics. He worked at the [[Ministry of Public Services]] as an engineer in charge of northern railway development from 1881 to 1885. He eventually became chief engineer of the Corps de Mines in 1893 and inspector general in 1910.

He never fully abandoned his mining career to mathematics. He worked at the Ministry of Public Services as an engineer in charge of northern railway development from 1881 to 1885. He eventually became chief engineer of the Corps de Mines in 1893 and inspector general in 1910.

他从未完全放弃他的采矿生涯数学。1881年至1885年，他在公共服务部担任工程师，负责北方铁路的发展。他最终在1893年成为矿业公司的总工程师，1910年成为监察长。



Beginning in 1881 and for the rest of his career, he taught at the University of Paris (the [[University of Paris|Sorbonne]]). He was initially appointed as the ''maître de conférences d'analyse'' (associate professor of analysis).<ref>Sageret, 1911</ref> Eventually, he held the chairs of Physical and Experimental Mechanics, Mathematical Physics and Theory of Probability,<ref>{{cite book|first =Laurent|last= Mazliak|chapter= Poincaré’s Odds |title = Poincaré 1912-2012 : Poincaré Seminar 2012|editor1-first= B.|editor1-last= Duplantier |editor2-first= V.|editor2-last= Rivasseau|volume = 67 |series = Progress in Mathematical Physics|publisher = Springer|isbn = 9783034808347|location = Basel|page = 150|url = https://books.google.com/books?id=njNpBQAAQBAJ|date= 14 November 2014}}</ref> and Celestial Mechanics and Astronomy.

Beginning in 1881 and for the rest of his career, he taught at the University of Paris (the Sorbonne). He was initially appointed as the maître de conférences d'analyse (associate professor of analysis). Eventually, he held the chairs of Physical and Experimental Mechanics, Mathematical Physics and Theory of Probability, and Celestial Mechanics and Astronomy.

从1881年开始，他在巴黎大学(索邦大学)教书，直到他的职业生涯结束。他最初被任命为分析师(分析学副教授)。最终，他获得了物理力学和实验力学、数学物理学和概率论、天体力学和天文学的学位。



In 1887, at the young age of 32, Poincaré was elected to the [[French Academy of Sciences]]. He became its president in 1906, and was elected to the [[Académie française]] on 5 March 1908.

In 1887, at the young age of 32, Poincaré was elected to the French Academy of Sciences. He became its president in 1906, and was elected to the Académie française on 5 March 1908.

1887年，32岁的庞加莱当选为法国科学院院士。他于1906年成为法兰西学术院主席，并于1908年3月5日当选为议员。



In 1887, he won [[Oscar II of Sweden|Oscar II, King of Sweden]]'s mathematical competition for a resolution of the [[three-body problem]] concerning the free motion of multiple orbiting bodies. (See [[#Three-body problem|three-body problem]] section below.)

In 1887, he won Oscar II, King of Sweden's mathematical competition for a resolution of the three-body problem concerning the free motion of multiple orbiting bodies. (See three-body problem section below.)

在1887年，他赢得了奥斯卡二世，瑞典国王的数学竞赛，以获得关于多轨道天体自由运动的三体。(见下面的三体问题。)



[[File:Poincaré gravestone.jpg|upright|thumb|The Poincaré family grave at the [[Cimetière du Montparnasse]]]]

The Poincaré family grave at the [[Cimetière du Montparnasse]]

庞加莱家族在[蒙帕纳斯公墓]的坟墓

In 1893, Poincaré joined the French [[Bureau des Longitudes]], which engaged him in the [[Clock synchronization|synchronisation of time]] around the world. In 1897 Poincaré backed an unsuccessful proposal for the [[Decimal degrees|decimalisation of circular measure]], and hence time and [[longitude]].<ref>see Galison 2003</ref> It was this post which led him to consider the question of establishing international time zones and the synchronisation of time between bodies in relative motion. (See [[#Work on relativity|work on relativity]] section below.)

In 1893, Poincaré joined the French Bureau des Longitudes, which engaged him in the synchronisation of time around the world. In 1897 Poincaré backed an unsuccessful proposal for the decimalisation of circular measure, and hence time and longitude. It was this post which led him to consider the question of establishing international time zones and the synchronisation of time between bodies in relative motion. (See work on relativity section below.)

1893年，庞加莱加入了法国法国经度管理局，参与了世界各地的时间同步研究。1897年庞加莱支持了一个不成功的提议，即圆形测量的十进制化，因此时间和经度也是如此。正是这个职位使他考虑建立国际时区和相对运动的物体之间时间同步的问题。(见下面相对论部分的工作。)



In 1899, and again more successfully in 1904, he intervened in the trials of [[Alfred Dreyfus]]. He attacked the spurious scientific claims of some of the evidence brought against Dreyfus, who was a Jewish officer in the French army charged with treason by colleagues.

In 1899, and again more successfully in 1904, he intervened in the trials of Alfred Dreyfus. He attacked the spurious scientific claims of some of the evidence brought against Dreyfus, who was a Jewish officer in the French army charged with treason by colleagues.

1899年，更成功的是1904年，他介入了对阿尔弗雷德 · 德雷福斯的审判。他抨击了一些针对德雷福斯的虚假科学证据，德雷福斯是法国军队中一名被同事指控犯有叛国罪的犹太军官。



Poincaré was the President of the [[Société astronomique de France|Société Astronomique de France (SAF)]], the French astronomical society, from 1901 to 1903.<ref name=BSAF1911>[http://gallica.bnf.fr/ark:/12148/bpt6k9626551q/f616.item ''Bulletin de la Société astronomique de France'', 1911, vol. 25, pp. 581–586]</ref>

Poincaré was the President of the Société Astronomique de France (SAF), the French astronomical society, from 1901 to 1903.

1901年至1903年，庞加莱是法国天文学会---- 法国天文学会天文学联合会的主席。



====Students====

Poincaré had two notable doctoral students at the University of Paris, [[Louis Bachelier]] (1900) and [[Dimitrie Pompeiu]] (1905).<ref>[http://www.genealogy.ams.org/id.php?id=34227 Mathematics Genealogy Project] {{Webarchive|url=https://web.archive.org/web/20071005011853/http://www.genealogy.ams.org/id.php?id=34227 |date=5 October 2007 }} North Dakota State University. Retrieved April 2008.</ref>

Poincaré had two notable doctoral students at the University of Paris, Louis Bachelier (1900) and Dimitrie Pompeiu (1905).

庞加莱在巴黎大学有两个著名的博士生，路易斯 · 巴切利耶(1900年)和迪米特里 · 庞佩尤(1905年)。



=== Death ===



In 1912, Poincaré underwent surgery for a [[prostate]] problem and subsequently died from an [[embolism]] on 17 July 1912, in Paris. He was 58 years of age. He is buried in the Poincaré family vault in the [[Cimetière du Montparnasse|Cemetery of Montparnasse]], Paris.

In 1912, Poincaré underwent surgery for a prostate problem and subsequently died from an embolism on 17 July 1912, in Paris. He was 58 years of age. He is buried in the Poincaré family vault in the Cemetery of Montparnasse, Paris.

1912年，庞加莱因前列腺问题接受了手术，随后于1912年7月17日在巴黎死于栓塞。他当时58岁。他被安葬在蒙帕尔纳斯公墓的庞加莱家族墓穴里。



A former French Minister of Education, [[Claude Allègre]], proposed in 2004 that Poincaré be reburied in the [[Panthéon, Paris|Panthéon]] in Paris, which is reserved for French citizens only of the highest honour.<ref>[https://web.archive.org/web/20041127160356/http://www.lexpress.fr/idees/tribunes/dossier/allegre/dossier.asp?ida=430274 Lorentz, Poincaré et Einstein]</ref>

A former French Minister of Education, Claude Allègre, proposed in 2004 that Poincaré be reburied in the Panthéon in Paris, which is reserved for French citizens only of the highest honour.

法国前教育部长克劳德 · 阿莱格雷在2004年提议将庞加莱重新安葬在巴黎的先贤祠，那里只为获得最高荣誉的法国公民保留。



==Work==



===Summary===

Poincaré made many contributions to different fields of pure and applied mathematics such as: [[celestial mechanics]], [[fluid mechanics]], [[optics]], electricity, [[telegraphy]], [[capillarity]], [[Elasticity (physics)|elasticity]], [[thermodynamics]], [[potential theory]], [[Quantum mechanics|quantum theory]], [[theory of relativity]] and [[physical cosmology]].

Poincaré made many contributions to different fields of pure and applied mathematics such as: celestial mechanics, fluid mechanics, optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory, theory of relativity and physical cosmology.

在纯数学和应用数学的不同领域做出了很多贡献，例如: 天体力学、流体力学、光学、电学、电报学、毛细现象、弹性力学、热力学、势论、量子理论、相对论和物理宇宙学。



He was also a populariser of mathematics and physics and wrote several books for the lay public.

He was also a populariser of mathematics and physics and wrote several books for the lay public.

他还是数学和物理的普及者，并为普通大众写了几本书。



Among the specific topics he contributed to are the following:

Among the specific topics he contributed to are the following:

他提出的具体主题包括:

*[[algebraic topology]]

*[[several complex variables|the theory of analytic functions of several complex variables]]

*[[abelian variety|the theory of abelian functions]]

*[[algebraic geometry]]

*the [[Poincaré conjecture]], proven in 2003 by [[Grigori Perelman]].

*[[Poincaré recurrence theorem]]

*[[hyperbolic geometry]]

*[[number theory]]

*the [[three-body problem]]

*[[diophantine equation|the theory of diophantine equations]]

*[[electromagnetism]]

*[[Special relativity|the special theory of relativity]]

*the [[fundamental group]]

*In the field of [[differential equations]] Poincaré has given many results that are critical for the qualitative theory of differential equations, for example the [[Poincaré homology sphere|Poincaré sphere]] and the [[Poincaré map]].

*Poincaré on "everybody's belief" in the [[q:Henri Poincaré|''Normal Law of Errors'']] (see [[normal distribution]] for an account of that "law")

*Published an influential paper providing a novel mathematical argument in support of [[quantum mechanics]].<ref name=McCormmach>{{Citation | last = McCormmach | first = Russell | title = Henri Poincaré and the Quantum Theory | journal = Isis | volume = 58 | issue = 1 | pages = 37–55 | date =Spring 1967 | doi =10.1086/350182| s2cid = 120934561 }}</ref><ref name=Irons>{{Citation | last = Irons | first = F. E. | title = Poincaré's 1911–12 proof of quantum discontinuity interpreted as applying to atoms | journal = American Journal of Physics | volume = 69 | issue = 8 | pages = 879–884 | date = August 2001 | doi =10.1119/1.1356056 |bibcode = 2001AmJPh..69..879I }}</ref>



===Three-body problem===

The problem of finding the general solution to the motion of more than two orbiting bodies in the solar system had eluded mathematicians since [[Isaac Newton|Newton's]] time. This was known originally as the three-body problem and later the [[n-body problem|''n''-body problem]], where ''n'' is any number of more than two orbiting bodies. The ''n''-body solution was considered very important and challenging at the close of the 19th century. Indeed, in 1887, in honour of his 60th birthday, [[Oscar II of Sweden|Oscar II, King of Sweden]], advised by [[Gösta Mittag-Leffler]], established a prize for anyone who could find the solution to the problem. The announcement was quite specific:

The problem of finding the general solution to the motion of more than two orbiting bodies in the solar system had eluded mathematicians since Newton's time. This was known originally as the three-body problem and later the n-body problem, where n is any number of more than two orbiting bodies. The n-body solution was considered very important and challenging at the close of the 19th century. Indeed, in 1887, in honour of his 60th birthday, Oscar II, King of Sweden, advised by Gösta Mittag-Leffler, established a prize for anyone who could find the solution to the problem. The announcement was quite specific:

自从牛顿时代以来，数学家们就一直没有解决太阳系中两个以上轨道天体运动的一般解的问题。这个问题最初被称为三体问题，后来又被称为 n 体问题，其中 n 是任意数量的两个以上的轨道天体。在19世纪末，n 体解被认为是非常重要和具有挑战性的。事实上，在1887年，为了庆祝他的60岁生日，瑞典国王奥斯卡二世在哥斯塔·米塔-列夫勒的建议下，设立了一个奖项，奖励任何能够找到解决问题的方法的人。声明非常具体:



<blockquote>Given a system of arbitrarily many mass points that attract each according to [[Newton's law of universal gravitation|Newton's law]], under the assumption that no two points ever collide, try to find a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of whose values the series [[uniform convergence|converges uniformly]].</blockquote>

<blockquote>Given a system of arbitrarily many mass points that attract each according to Newton's law, under the assumption that no two points ever collide, try to find a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of whose values the series converges uniformly.</blockquote>

给定一个由任意多个质点组成的系统，这些质点根据牛顿定律相互吸引，在假设没有两个点发生碰撞的情况下，试图找出每个点的坐标在一个变量中作为一个级数的表示，这个级数是某个已知的时间函数，对于所有这些值，这个级数均一致收敛。</blockquote >



In case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was finally awarded to Poincaré, even though he did not solve the original problem. One of the judges, the distinguished [[Karl Weierstrass]], said, ''"This work cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that its publication will inaugurate a new era in the history of celestial mechanics."'' (The first version of his contribution even contained a serious error; for details see the article by Diacu<ref name=diacu>{{Citation

In case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was finally awarded to Poincaré, even though he did not solve the original problem. One of the judges, the distinguished Karl Weierstrass, said, "This work cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that its publication will inaugurate a new era in the history of celestial mechanics." (The first version of his contribution even contained a serious error; for details see the article by Diacu and the book by Barrow-Green). The version finally printed contained many important ideas which led to the theory of chaos. The problem as stated originally was finally solved by Karl F. Sundman for n&nbsp;=&nbsp;3 in 1912 and was generalised to the case of n&nbsp;>&nbsp;3 bodies by Qiudong Wang in the 1990s.

如果这个问题无法解决，任何其他对经典力学的重要贡献都会被认为是值得的。尽管 Poincaré 没有解决最初的问题，但他最终获得了诺贝尔和平奖。其中一位评委，尊敬的卡尔·魏尔斯特拉斯，说，“这项工作确实不能被认为是提供了提议的问题的完整解决方案，但它是如此重要，它的出版将开创一个新的时代，在天体力学的历史。”(他的贡献的第一个版本甚至包含了一个严重的错误; 详情见 Diacu 的文章和 Barrow-Green 的书)。最后印刷出来的版本包含了许多重要的思想，这些思想导致了混沌理论的产生。1912年，Karl f. Sundman 最终解决了 n = 3的问题，1990年代，王将其推广到 n > 3具尸体的情况。

| last=Diacu|first= Florin | year=1996 | title=The solution of the ''n''-body Problem | journal=The Mathematical Intelligencer | volume =18 | pages =66–70 | doi=10.1007/BF03024313

| issue=3|s2cid= 119728316 }}</ref> and the book by [[June Barrow-Green|Barrow-Green]]<ref>{{Cite book|title=Poincaré and the three body problem|title-link= Poincaré and the Three-Body Problem |last=Barrow-Green|first=June|publisher=[[American Mathematical Society]]|year=1997|isbn=978-0821803677|location=Providence, RI|series=History of Mathematics|volume=11|pages=|oclc=34357985}}</ref>). The version finally printed<ref>{{Cite book|title=The three-body problem and the equations of dynamics: Poincaré's foundational work on dynamical systems theory|last=Poincaré|first=J. Henri|publisher=Springer International Publishing|others=Popp, Bruce D. (Translator)|year=2017|isbn=9783319528984|location=Cham, Switzerland|pages=|oclc=987302273}}</ref> contained many important ideas which led to the [[chaos theory|theory of chaos]]. The problem as stated originally was finally solved by [[Karl F. Sundman]] for ''n''&nbsp;=&nbsp;3 in 1912 and was generalised to the case of ''n''&nbsp;>&nbsp;3 bodies by [[Qiudong Wang]] in the 1990s.



[[Marie Curie and Poincaré talk at the 1911 Solvay Conference]]

[玛丽 · 居里和庞加莱在1911年索尔维会议大会上的演讲]

===Work on relativity===

[[Image:Curie and Poincare 1911 Solvay.jpg|thumb|right|[[Marie Curie]] and Poincaré talk at the 1911 [[Solvay Conference]]]]

{{main|Lorentz ether theory|History of special relativity}}



Poincaré's work at the Bureau des Longitudes on establishing international time zones led him to consider how clocks at rest on the Earth, which would be moving at different speeds relative to absolute space (or the "luminiferous aether"), could be synchronised. At the same time Dutch theorist Hendrik Lorentz was developing Maxwell's theory into a theory of the motion of charged particles ("electrons" or "ions"), and their interaction with radiation. In 1895 Lorentz had introduced an auxiliary quantity (without physical interpretation) called "local time" <math>t^\prime = t-v x/c^2 \,</math>

庞加莱在法国经度管理局关于建立国际时区的工作使他思考如何使地球上静止的时钟以不同的速度相对于绝对空间(或称为“以太时间”)进行同步。与此同时，荷兰理论家亨德里克 · 洛伦兹正在将麦克斯韦理论发展成一个关于带电粒子(“电子”或“离子”)运动及其与辐射相互作用的理论。1895年，洛伦兹引入了一个辅助量(没有物理解释) ，叫做“本地时间” t ^ prime = t-v x/c ^ 2

====Local time====

and introduced the hypothesis of length contraction to explain the failure of optical and electrical experiments to detect motion relative to the aether (see Michelson–Morley experiment).

并且引入了长度收缩假说来解释光学和电学实验相对于以太探测运动的失败(见 Michelson-Morley 实验)。

Poincaré's work at the Bureau des Longitudes on establishing international time zones led him to consider how clocks at rest on the Earth, which would be moving at different speeds relative to absolute space (or the "[[luminiferous aether]]"), could be synchronised. At the same time Dutch theorist [[Hendrik Lorentz]] was developing Maxwell's theory into a theory of the motion of charged particles ("electrons" or "ions"), and their interaction with radiation. In 1895 Lorentz had introduced an auxiliary quantity (without physical interpretation) called "local time" <math>t^\prime = t-v x/c^2 \,</math><ref>{{Citation|title=A broader view of relativity: general implications of Lorentz and Poincaré invariance|volume=10|first1=Jong-Ping|last1=Hsu|first2=Leonardo|last2=Hsu|publisher=World Scientific|year=2006|isbn=978-981-256-651-5|page=37

Poincaré was a constant interpreter (and sometimes friendly critic) of Lorentz's theory. Poincaré as a philosopher was interested in the "deeper meaning". Thus he interpreted Lorentz's theory and in so doing he came up with many insights that are now associated with special relativity. In The Measure of Time (1898), Poincaré said, "

庞加莱是洛伦兹理论的不断解释者(有时也是友好的批评者)。作为一个哲学家，庞加莱对“更深层的意义”很感兴趣。因此，他解释了 Lorentz 的理论，并由此提出了许多现在与狭义相对论有关的见解。在《时间的度量》(1898)中，庞加莱说:

|url=https://books.google.com/books?id=amLqckyrvUwC}}, [https://books.google.com/books?id=amLqckyrvUwC&pg=PA37 Section A5a, p 37]</ref>

A little reflection is sufficient to understand that all these affirmations have by themselves no meaning. They can have one only as the result of a convention." He also argued that scientists have to set the constancy of the speed of light as a postulate to give physical theories the simplest form.

稍加思考就足以理解所有这些自我肯定本身没有任何意义。他们只能根据惯例生一个。”他还认为，科学家必须把光速的恒定性作为一个假设，才能给物理理论提供最简单的形式。

and introduced the hypothesis of [[length contraction]] to explain the failure of optical and electrical experiments to detect motion relative to the aether (see [[Michelson–Morley experiment]]).<ref>{{Citation

Based on these assumptions he discussed in 1900 Lorentz's "wonderful invention" of local time and remarked that it arose when moving clocks are synchronised by exchanging light signals assumed to travel with the same speed in both directions in a moving frame.

基于这些假设，他在1900年讨论了洛伦兹关于当地时间的“奇妙发明” ，并指出，当移动的时钟通过交换假定在移动的框架中以相同速度向两个方向移动的光信号而实现同步时，就产生了这一假设。

| last=Lorentz|first= Hendrik A. | authorlink=Hendrik Lorentz| year=1895 | title=Versuch einer theorie der electrischen und optischen erscheinungen in bewegten Kõrpern | place =Leiden| publisher=E.J. Brill| title-link=s:de:Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern }}</ref>

Poincaré was a constant interpreter (and sometimes friendly critic) of Lorentz's theory. Poincaré as a philosopher was interested in the "deeper meaning". Thus he interpreted Lorentz's theory and in so doing he came up with many insights that are now associated with special relativity. In [[s:The Measure of Time|The Measure of Time]] (1898), Poincaré said, "

A little reflection is sufficient to understand that all these affirmations have by themselves no meaning. They can have one only as the result of a convention." He also argued that scientists have to set the constancy of the speed of light as a [[postulate]] to give physical theories the simplest form.<ref>{{Citation

| last=Poincaré|first= Henri | year=1898 | title=The Measure of Time | journal=Revue de Métaphysique et de Morale | volume =6 | pages =1–13| title-link=s:The Measure of Time }}</ref>

Based on these assumptions he discussed in 1900 Lorentz's "wonderful invention" of local time and remarked that it arose when moving clocks are synchronised by exchanging light signals assumed to travel with the same speed in both directions in a moving frame.<ref name=action>{{Citation

In 1881 Poincaré described hyperbolic geometry in terms of the hyperboloid model, formulating transformations leaving invariant the Lorentz interval <math>x^2+y^2-z^2=-1</math>, which makes them mathematically equivalent to the Lorentz transformations in 2+1 dimensions. In addition, Poincaré's other models of hyperbolic geometry (Poincaré disk model, Poincaré half-plane model) as well as the Beltrami–Klein model can be related to the relativistic velocity space (see Gyrovector space).

在1881年庞加莱用双曲几何描述了双曲面模型变换，公式化的变换保持洛伦兹区间不变。此外，poincaré 的其他双曲几何模型(庞加莱圆盘模型，庞加莱半平面模型)以及 Beltrami-Klein 模型可以与相对论速度空间相关(见回旋向量空间)。

| last=Poincaré|first= Henri | year=1900 | title=La théorie de Lorentz et le principe de réaction | journal=Archives Néerlandaises des Sciences Exactes et Naturelles | volume =5 | pages =252–278| title-link=s:fr:La théorie de Lorentz et le principe de réaction }}. See also the [http://www.physicsinsights.org/poincare-1900.pdf English translation]</ref>



In 1892 Poincaré developed a mathematical theory of light including polarization. His vision of the action of polarizers and retarders, acting on a sphere representing polarized states, is called the Poincaré sphere. It was shown that the Poincaré sphere possesses an underlying Lorentzian symmetry, by which it can be used as a geometrical representation of Lorentz transformations and velocity additions.

1892年庞加莱发展了包括偏振在内的光的数学理论。他关于偏振器和延迟器作用于代表极化状态的球体的观点称为庞加莱球。证明了庞加莱球具有一个基本的洛伦兹对称性，可以作为洛伦兹变换和速度加法的几何表示。

====Principle of relativity and Lorentz transformations====

{{Further|History of Lorentz transformations#Poincare|History of Lorentz transformations#Poincare3|label1=History of Lorentz transformations - Poincaré (1881)|label2=History of Lorentz transformations - Poincaré (1905)}}

He discussed the "principle of relative motion" in two papers in 1900

他在1900年的两篇论文中讨论了“相对运动原理”



and named it the principle of relativity in 1904, according to which no physical experiment can discriminate between a state of uniform motion and a state of rest.

并在1904年将其命名为相对性原理，根据这一理论，没有任何物理实验能够区分匀速运动状态和静止状态。

In 1881 Poincaré described [[hyperbolic geometry]] in terms of the [[hyperboloid model]], formulating transformations leaving invariant the [[Lorentz interval]] <math>x^2+y^2-z^2=-1</math>, which makes them mathematically equivalent to the Lorentz transformations in 2+1 dimensions.<ref>{{Cite journal|author=Poincaré, H.|year=1881|title=Sur les applications de la géométrie non-euclidienne à la théorie des formes quadratiques|journal=Association Française Pour l'Avancement des Sciences|volume=10|pages=132–138|url=http://henripoincarepapers.univ-nantes.fr/chp/hp-pdf/hp1881af.pdf}}{{Dead link|date=June 2020 |bot=InternetArchiveBot |fix-attempted=yes }}</ref><ref>{{Cite journal|author=Reynolds, W. F.|year=1993|title=Hyperbolic geometry on a hyperboloid|journal=The American Mathematical Monthly|volume=100|issue=5|pages=442–455|jstor=2324297|doi=10.1080/00029890.1993.11990430}}</ref> In addition, Poincaré's other models of hyperbolic geometry ([[Poincaré disk model]], [[Poincaré half-plane model]]) as well as the [[Beltrami–Klein model]] can be related to the relativistic velocity space (see [[Gyrovector space]]).

In 1905 Poincaré wrote to Lorentz about Lorentz's paper of 1904, which Poincaré described as a "paper of supreme importance." In this letter he pointed out an error Lorentz had made when he had applied his transformation to one of Maxwell's equations, that for charge-occupied space, and also questioned the time dilation factor given by Lorentz.

1905年庞加莱写信给洛伦兹，谈到他1904年的论文，庞加莱称之为“极其重要的论文”在这封信中，他指出了洛伦兹在对麦克斯韦方程组中的一个电荷占据空间进行变换时所犯的一个错误，并对洛伦兹给出的时间膨胀因子提出了质疑。



In a second letter to Lorentz, Poincaré gave his own reason why Lorentz's time dilation factor was indeed correct after all—it was necessary to make the Lorentz transformation form a group—and he gave what is now known as the relativistic velocity-addition law.

在写给洛伦兹的第二封信中，庞加莱给出了他自己的理由，为什么洛伦兹的时间膨胀因子终究是正确的ーー把洛伦兹变换变成一个群是必要的ーー他还给出了现在已知的相对论速度加和定律。

In 1892 Poincaré developed a mathematical theory of light including [[polarization (waves)|polarization]]. His vision of the action of polarizers and retarders, acting on a sphere representing polarized states, is called the [[Poincaré sphere (optics)|Poincaré sphere]].<ref>{{Cite book|author=Poincaré, H. |year=1892|title=Théorie mathématique de la lumière II|location=Paris|publisher=Georges Carré|chapter-url=https://archive.org/details/thoriemathma00poin|chapter=Chapitre XII: Polarisation rotatoire}}</ref> It was shown that the Poincaré sphere possesses an underlying Lorentzian symmetry, by which it can be used as a geometrical representation of Lorentz transformations and velocity additions.<ref>{{Cite journal|author=Tudor, T.|year=2018|title=Lorentz Transformation, Poincaré Vectors and Poincaré Sphere in Various Branches of Physics|journal=Symmetry|volume=10|issue=3|pages=52|doi=10.3390/sym10030052|doi-access=free}}</ref>

Poincaré later delivered a paper at the meeting of the Academy of Sciences in Paris on 5 June 1905 in which these issues were addressed. In the published version of that he wrote:

后来，庞加莱在1905年6月5日于巴黎举行的科学院会议上发表了一篇论文，论述了这些问题。在出版的版本中，他写道:



He discussed the "principle of relative motion" in two papers in 1900<ref name=action /><ref>{{Citation

<blockquote>The essential point, established by Lorentz, is that the equations of the electromagnetic field are not altered by a certain transformation (which I will call by the name of Lorentz) of the form:

洛伦兹建立的基本观点是，电磁场的方程式不会因为某种形式的变换而改变(我称之为洛伦兹) :

| author=Poincaré, H. | year=1900 | title= Les relations entre la physique expérimentale et la physique mathématique | journal=Revue Générale des Sciences Pures et Appliquées | volume =11 | pages =1163–1175 | url=http://gallica.bnf.fr/ark:/12148/bpt6k17075r/f1167.table}}. Reprinted in "Science and Hypothesis", Ch. 9–10.</ref>

<math>x^\prime = k\ell\left(x + \varepsilon t\right)\!,\;t^\prime = k\ell\left(t + \varepsilon x\right)\!,\;y^\prime = \ell y,\;z^\prime = \ell z,\;k = 1/\sqrt{1-\varepsilon^2}.</math></blockquote>

X ^ prime = k ell left (x + varepsilon t right) ! ; t ^ prime = k ell left (t + varepsilon x right) ! ; y ^ prime = ell y; z ^ prime = ell z; k = 1/sqrt {1-varepsilon ^ 2} </math > </blockquote >

and named it the [[principle of relativity]] in 1904, according to which no physical experiment can discriminate between a state of uniform motion and a state of rest.<ref name=louis>{{Citation|author=Poincaré, Henri|year=1913|chapter=[[s:The Principles of Mathematical Physics|The Principles of Mathematical Physics]]|title=The Foundations of Science (The Value of Science)|pages=297–320|publisher=Science Press|place=New York|postscript=; article translated from 1904 original}} available in [https://books.google.com/books/about/The_Foundations_of_Science.html?id=mBvNabP35zoC&pg=PA297 online chapter from 1913 book]</ref>

In 1905 Poincaré wrote to Lorentz about Lorentz's paper of 1904, which Poincaré described as a "paper of supreme importance." In this letter he pointed out an error Lorentz had made when he had applied his transformation to one of Maxwell's equations, that for charge-occupied space, and also questioned the time dilation factor given by Lorentz.<ref name="univ-nantes">

and showed that the arbitrary function <math>\ell\left(\varepsilon\right)</math> must be unity for all <math>\varepsilon</math> (Lorentz had set <math>\ell = 1</math> by a different argument) to make the transformations form a group. In an enlarged version of the paper that appeared in 1906 Poincaré pointed out that the combination <math>x^2+ y^2+ z^2- c^2t^2</math> is invariant. He noted that a Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing <math>ct\sqrt{-1}</math> as a fourth imaginary coordinate, and he used an early form of four-vectors. Poincaré expressed a lack of interest in a four-dimensional reformulation of his new mechanics in 1907, because in his opinion the translation of physics into the language of four-dimensional geometry would entail too much effort for limited profit. So it was Hermann Minkowski who worked out the consequences of this notion in 1907.

证明了任意函数“左”(右)必为“数”(Lorentz 用另一个论点设置“数”)的整数，从而使变换形成一个组。在1906年庞加莱论文的放大版本中，他指出组合 x ^ 2 + y ^ 2 + z ^ 2-c ^ 2 </math > 是不变的。他指出，洛伦兹变换是通过引入 < math > ct sqrt {-1} </math > 作为第四个虚数坐标，仅仅是在原点四维上的一个旋转，并且他使用了早期形式的四向量。庞加莱在1907年对其新力学的四维重新表述缺乏兴趣，因为他认为，将物理学翻译成四维几何学的语言将需要为有限的利润付出太多的努力。因此，在1907年，赫尔曼·闵可夫斯基 · 马丁发现了这个概念的后果。

{{Citation | author=Poincaré, H. | year=2007 | editor=Walter, S. A. | contribution= 38.3, Poincaré to H. A. Lorentz, May 1905 | title=La correspondance entre Henri Poincaré et les physiciens, chimistes, et ingénieurs |pages=255–257 |place=Basel | publisher=Birkhäuser|contribution-url=http://henripoincarepapers.univ-nantes.fr/chp/text/lorentz3.html}}</ref>

In a second letter to Lorentz, Poincaré gave his own reason why Lorentz's time dilation factor was indeed correct after all—it was necessary to make the Lorentz transformation form a group—and he gave what is now known as the relativistic velocity-addition law.<ref name="univ-nantes2">{{Citation | author=Poincaré, H. | year=2007 | editor=Walter, S. A. | contribution= 38.4, Poincaré to H. A. Lorentz, May 1905 | title=La correspondance entre Henri Poincaré et les physiciens, chimistes, et ingénieurs |pages=257–258 |place=Basel | publisher=Birkhäuser|contribution-url=http://henripoincarepapers.univ-nantes.fr/chp/text/lorentz4.html}}</ref>

Poincaré later delivered a paper at the meeting of the Academy of Sciences in Paris on 5 June 1905 in which these issues were addressed. In the published version of that he wrote:<ref name="1905 paper">[http://www.academie-sciences.fr/pdf/dossiers/Poincare/Poincare_pdf/Poincare_CR1905.pdf] (PDF) Membres de l'Académie des sciences depuis sa création : Henri Poincare. Sur la dynamique de l' electron. Note de H. Poincaré. C.R. T.140 (1905) 1504–1508.</ref>

Like others before, Poincaré (1900) discovered a relation between mass and electromagnetic energy. While studying the conflict between the action/reaction principle and Lorentz ether theory, he tried to determine whether the center of gravity still moves with a uniform velocity when electromagnetic fields are included. the possibility that energy carries mass and criticized the ether solution to compensate the above-mentioned problems:

像其他人一样，庞加莱(1900)发现了质量和电磁能量之间的关系。在研究作用力/反作用力原理和洛伦兹理论之间的冲突时，他试图确定当电磁场包括在内时，重心是否仍以均匀速度运动。能量携带质量并且批评以太解决方案来补偿上述问题的可能性:



<blockquote>The essential point, established by Lorentz, is that the equations of the electromagnetic field are not altered by a certain transformation (which I will call by the name of Lorentz) of the form:

::<math>x^\prime = k\ell\left(x + \varepsilon t\right)\!,\;t^\prime = k\ell\left(t + \varepsilon x\right)\!,\;y^\prime = \ell y,\;z^\prime = \ell z,\;k = 1/\sqrt{1-\varepsilon^2}.</math></blockquote>



He also discussed two other unexplained effects: (1) non-conservation of mass implied by Lorentz's variable mass <math>\gamma m</math>, Abraham's theory of variable mass and Kaufmann's experiments on the mass of fast moving electrons and (2) the non-conservation of energy in the radium experiments of Madame Curie.

他还讨论了另外两个无法解释的效应: (1)洛伦兹变质量理论暗示的质量不守恒，亚伯拉罕变质量理论和考夫曼关于快速运动电子质量的实验，以及(2)居里夫人镭实验中的能量不守恒。

and showed that the arbitrary function <math>\ell\left(\varepsilon\right)</math> must be unity for all <math>\varepsilon</math> (Lorentz had set <math>\ell = 1</math> by a different argument) to make the transformations form a group. In an enlarged version of the paper that appeared in 1906 Poincaré pointed out that the combination <math>x^2+ y^2+ z^2- c^2t^2</math> is [[Invariant (mathematics)|invariant]]. He noted that a Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing <math>ct\sqrt{-1}</math> as a fourth imaginary coordinate, and he used an early form of [[four-vector]]s.<ref name=long>{{Citation

| author=Poincaré, H. | year=1906 | title=Sur la dynamique de l'électron (On the Dynamics of the Electron) | journal=Rendiconti del Circolo Matematico Rendiconti del Circolo di Palermo | volume =21 | pages =129–176

It was Albert Einstein's concept of mass–energy equivalence (1905) that a body losing energy as radiation or heat was losing mass of amount m&nbsp;=&nbsp;E/c<sup>2</sup> that resolved Poincaré's paradox, without using any compensating mechanism within the ether. The Hertzian oscillator loses mass in the emission process, and momentum is conserved in any frame. However, concerning Poincaré's solution of the Center of Gravity problem, Einstein noted that Poincaré's formulation and his own from 1906 were mathematically equivalent.

阿尔伯特 · 爱因斯坦(Albert Einstein)的质能等效(mass-energy equivalence，1905)概念解决了庞加莱悖论(poincaré 佯谬) ，而没有使用以太中的任何补偿机制。赫兹振子在发射过程中失去了质量，动量在任何一个框架中都是守恒的。然而，关于庞加莱的重心问题的解决方案，爱因斯坦指出，庞加莱的公式和他自己1906年的公式在数学上是等价的。

| doi=10.1007/BF03013466| bibcode=1906RCMP...21..129P| hdl=2027/uiug.30112063899089 | s2cid=120211823 | url=https://zenodo.org/record/1428444| hdl-access=free }} (Wikisource translation)</ref> Poincaré expressed a lack of interest in a four-dimensional reformulation of his new mechanics in 1907, because in his opinion the translation of physics into the language of four-dimensional geometry would entail too much effort for limited profit.<ref>Walter (2007), Secondary sources on relativity</ref> So it was [[Hermann Minkowski]] who worked out the consequences of this notion in 1907.



====Mass–energy relation====

In 1905 Henri Poincaré first proposed gravitational waves (ondes gravifiques) emanating from a body and propagating at the speed of light. In public, Einstein acknowledged Poincaré posthumously in the text of a lecture in 1921 called Geometrie und Erfahrung in connection with non-Euclidean geometry, but not in connection with special relativity. A few years before his death, Einstein commented on Poincaré as being one of the pioneers of relativity, saying "Lorentz had already recognised that the transformation named after him is essential for the analysis of Maxwell's equations, and Poincaré deepened this insight still further ...."

1905年，Henri poincaré 首次提出引力波(ondes gravitfiques) ，它从物体中发射出来，以光速传播。在公开场合，他在1921年的一次名为几何和历史的演讲中承认了庞加莱，这次演讲与非欧几里得几何有关，但与狭义相对论无关。在他去世前几年，爱因斯坦评价庞加莱是相对论的先驱之一，他说: “洛伦兹已经认识到，以他的名字命名的变换对于分析麦克斯韦方程组至关重要，而庞加莱进一步深化了这一认识... ... ”

Like [[Mass–energy equivalence#Electromagnetic rest mass|others]] before, Poincaré (1900) discovered a relation between mass and electromagnetic energy. While studying the conflict between the [[Newton's laws of motion|action/reaction principle]] and [[Lorentz ether theory]], he tried to determine whether the [[center of gravity]] still moves with a uniform velocity when electromagnetic fields are included.<ref name=action /> He noticed that the action/reaction principle does not hold for matter alone, but that the electromagnetic field has its own momentum. Poincaré concluded that the electromagnetic field energy of an electromagnetic wave behaves like a fictitious [[fluid]] (''fluide fictif'') with a mass density of ''E''/''c''<sup>2</sup>. If the [[center of mass frame]] is defined by both the mass of matter ''and'' the mass of the fictitious fluid, and if the fictitious fluid is indestructible—it's neither created or destroyed—then the motion of the center of mass frame remains uniform. But electromagnetic energy can be converted into other forms of energy. So Poincaré assumed that there exists a non-electric energy fluid at each point of space, into which electromagnetic energy can be transformed and which also carries a mass proportional to the energy. In this way, the motion of the center of mass remains uniform. Poincaré said that one should not be too surprised by these assumptions, since they are only mathematical fictions.



However, Poincaré's resolution led to a paradox when changing frames: if a Hertzian oscillator radiates in a certain direction, it will suffer a [[recoil]] from the inertia of the fictitious fluid. Poincaré performed a [[Lorentz boost]] (to order ''v''/''c'') to the frame of the moving source. He noted that energy conservation holds in both frames, but that the law of conservation of momentum is violated. This would allow [[perpetual motion]], a notion which he abhorred. The laws of nature would have to be different in the frames of reference, and the relativity principle would not hold. Therefore, he argued that also in this case there has to be another compensating mechanism in the ether.



Poincaré's work in the development of special relativity is well recognised, Poincaré developed a similar physical interpretation of local time and noticed the connection to signal velocity, but contrary to Einstein he continued to use the ether-concept in his papers and argued that clocks at rest in the ether show the "true" time, and moving clocks show the local time. So Poincaré tried to keep the relativity principle in accordance with classical concepts, while Einstein developed a mathematically equivalent kinematics based on the new physical concepts of the relativity of space and time.

庞加莱在发展狭义相对论方面的工作得到了广泛认可，庞加莱对当地时间进行了类似的物理解释，并注意到了与信号速度的联系，但与爱因斯坦相反，他在论文中继续使用以太的概念，认为静止在以太中的时钟显示“真实”的时间，而移动的时钟显示当地时间。因此庞加莱试图使相对论原理与经典概念保持一致，而爱因斯坦则基于空间和时间相对论的新物理概念，发展了一个数学等价的运动学。

Poincaré himself came back to this topic in his St. Louis lecture (1904).<ref name=louis /> This time (and later also in 1908) he rejected<ref>Miller 1981, Secondary sources on relativity</ref> the possibility that energy carries mass and criticized the ether solution to compensate the above-mentioned problems:



While this is the view of most historians, a minority go much further, such as E. T. Whittaker, who held that Poincaré and Lorentz were the true discoverers of relativity.

虽然这是大多数历史学家的观点，少数人走得更远，如惠特克，他认为，Poincaré 和洛伦兹是真正的发现者相对论。

{{quote|The apparatus will recoil as if it were a cannon and the projected energy a ball, and that contradicts the principle of Newton, since our present projectile has no mass; it is not matter, it is energy. [..] Shall we say that the space which separates the oscillator from the receiver and which the disturbance must traverse in passing from one to the other, is not empty, but is filled not only with ether, but with air, or even in inter-planetary space with some subtile, yet ponderable fluid; that this matter receives the shock, as does the receiver, at the moment the energy reaches it, and recoils, when the disturbance leaves it? That would save Newton's principle, but it is not true. If the energy during its propagation remained always attached to some material substratum, this matter would carry the light along with it and Fizeau has shown, at least for the air, that there is nothing of the kind. Michelson and Morley have since confirmed this. We might also suppose that the motions of matter proper were exactly compensated by those of the ether; but that would lead us to the same considerations as those made a moment ago. The principle, if thus interpreted, could explain anything, since whatever the visible motions we could imagine hypothetical motions to compensate them. But if it can explain anything, it will allow us to foretell nothing; it will not allow us to choose between the various possible hypotheses, since it explains everything in advance. It therefore becomes useless. }}



He also discussed two other unexplained effects: (1) non-conservation of mass implied by Lorentz's variable mass <math>\gamma m</math>, Abraham's theory of variable mass and [[Walter Kaufmann (physicist)|Kaufmann]]'s experiments on the mass of fast moving electrons and (2) the non-conservation of energy in the radium experiments of [[Madame Curie]].

Poincaré introduced group theory to physics, and was the first to study the group of Lorentz transformations. He also made major contributions to the theory of discrete groups and their representations.

庞加莱将群论引入物理学，并且是第一个研究洛伦兹变换群的人。他还对离散群及其表示理论作出了重大贡献。



<center>Topological transformation of the torus into a mug </center>

环面向杯子的拓扑变换

It was [[Albert Einstein]]'s concept of [[mass–energy equivalence]] (1905) that a body losing energy as radiation or heat was losing mass of amount ''m''&nbsp;=&nbsp;''E''/''c''<sup>2</sup> that resolved<ref name=darrigol>Darrigol 2005, Secondary sources on relativity</ref> Poincaré's paradox, without using any compensating mechanism within the ether.<ref>{{Citation|author=Einstein, A. |year=1905b |title=Ist die Trägheit eines Körpers von dessen Energieinhalt abhängig? |journal=Annalen der Physik |volume=18 |issue=13 |pages=639–643 |bibcode=1905AnP...323..639E |doi= 10.1002/andp.19053231314 |url=http://www.physik.uni-augsburg.de/annalen/history/papers/1905_18_639-641.pdf |archive-url=https://web.archive.org/web/20050124051500/http://www.physik.uni-augsburg.de/annalen/history/papers/1905_18_639-641.pdf |url-status=dead |archive-date=24 January 2005}}. See also [http://www.fourmilab.ch/etexts/einstein/specrel/www English translation].</ref> The Hertzian oscillator loses mass in the emission process, and momentum is conserved in any frame. However, concerning Poincaré's solution of the Center of Gravity problem, Einstein noted that Poincaré's formulation and his own from 1906 were mathematically equivalent.<ref>{{Citation|author=Einstein, A. |year=1906 |title=Das Prinzip von der Erhaltung der Schwerpunktsbewegung und die Trägheit der Energie |journal=Annalen der Physik |volume=20 |pages=627–633 |doi=10.1002/andp.19063250814 |issue=8 |bibcode=1906AnP...325..627E |url= http://www.physik.uni-augsburg.de/annalen/history/papers/1906_20_627-633.pdf |archive-url=https://web.archive.org/web/20060318060830/http://www.physik.uni-augsburg.de/annalen/history/papers/1906_20_627-633.pdf |url-status=dead |archive-date=18 March 2006}}</ref>



====Gravitational waves====

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.

这个主题是由 Felix Klein 在他的《爱尔兰根纲领(1872)中明确定义的: 任意连续变换的几何不变量，一种几何学。正如利斯廷所建议的那样，引入了术语“拓扑” ，而不是之前使用的“分析位置”。一些重要的概念是由 Enrico Betti 和波恩哈德·黎曼介绍的。但是对于任何维度的空间来说，这门科学的基础都是由庞加莱创造的。他的第一篇关于这个主题的文章发表于1894年。

In 1905 Henri Poincaré first proposed [[gravitational waves]] (''ondes gravifiques'') emanating from a body and propagating at the speed of light.<ref name="1905 paper" /> ''"Il importait d'examiner cette hypothèse de plus près et en particulier de rechercher quelles modifications elle nous obligerait à apporter aux lois de la gravitation. C'est ce que j'ai cherché à déterminer; j'ai été d'abord conduit à supposer que la propagation de la gravitation n'est pas instantanée, mais se fait avec la vitesse de la lumière."''



His research in geometry led to the abstract topological definition of homotopy and 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.

他对几何的研究导致了同伦和同调的抽象拓扑定义。他还首先介绍了组合拓扑的基本概念和不变量，如 Betti 数和基本群。证明了 n 维多面体的边数、顶点数和面数的一个公式(欧拉-庞加莱定理) ，给出了直观维数概念的第一个精确表达式。

====Poincaré and Einstein====

Einstein's first paper on relativity was published three months after Poincaré's short paper,<ref name="1905 paper" /> but before Poincaré's longer version.<ref name=long /> Einstein relied on the principle of relativity to derive the Lorentz transformations and used a similar clock synchronisation procedure ([[Einstein synchronisation]]) to the one that Poincaré (1900) had described, but Einstein's paper was remarkable in that it contained no references at all. Poincaré never acknowledged Einstein's work on [[special relativity]]. However, Einstein expressed sympathy with Poincaré's outlook obliquely in a letter to [[Hans Vaihinger]] on 3 May 1919, when Einstein considered Vaihinger's general outlook to be close to his own and Poincaré's to be close to Vaihinger's.<ref>{{cite book|series=The Collected Papers of Albert Einstein |url=http://einsteinpapers.press.princeton.edu/vol9-trans/52 |publisher=Princeton U.P. |accessdate=|volume = 9|title = The Berlin Years: Correspondence, January 1919-April 1920 (English translation supplement)|page = 30}} See also this letter, with commentary, in {{cite journal |last=Sass |first=Hans-Martin | authorlink = Hans-Martin Sass|date=1979 |title=Einstein über "wahre Kultur" und die Stellung der Geometrie im Wissenschaftssystem: Ein Brief Albert Einsteins an Hans Vaihinger vom Jahre 1919 |journal=[[Zeitschrift für allgemeine Wissenschaftstheorie]] |volume=10 |issue=2 |pages=316–319 |jstor=25170513 |language=de |doi=10.1007/bf01802352|s2cid=170178963 }}</ref> In public, Einstein acknowledged Poincaré posthumously in the text of a lecture in 1921 called ''Geometrie und Erfahrung'' in connection with [[non-Euclidean geometry]], but not in connection with special relativity. A few years before his death, Einstein commented on Poincaré as being one of the pioneers of relativity, saying "Lorentz had already recognised that the transformation named after him is essential for the analysis of Maxwell's equations, and Poincaré deepened this insight still further ...."<ref>Darrigol 2004, Secondary sources on relativity</ref>



<center> Chaotic motion in three-body problem (computer simulation).</center>

< 中心 > 三体问题的混乱运动(计算机模拟) </中心 >

====Assessments on Poincaré and relativity====

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.

庞加莱出版了两本经典专著《天体力学》(1892-1899)和《天体力学》(1905-1910)。其中，他成功地将他们的研究成果应用于三体运动问题，并详细研究了解的行为(频率、稳定性、渐近性等)。介绍了小参数方法、不动点、积分不变量、变分方程、渐近展开式的收敛性。将 Bruns (1887)的理论进行概括，poincaré 指出三体不可积。换句话说，三体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。他在这个领域的工作是自艾萨克 · 牛顿以来天体力学的第一个重大成就。

{{Further|History of special relativity|Relativity priority dispute}}

Poincaré's work in the development of special relativity is well recognised,<ref name=darrigol /> though most historians stress that despite many similarities with Einstein's work, the two had very different research agendas and interpretations of the work.<ref>Galison 2003 and Kragh 1999, Secondary sources on relativity</ref> Poincaré developed a similar physical interpretation of local time and noticed the connection to signal velocity, but contrary to Einstein he continued to use the ether-concept in his papers and argued that clocks at rest in the ether show the "true" time, and moving clocks show the local time. So Poincaré tried to keep the relativity principle in accordance with classical concepts, while Einstein developed a mathematically equivalent kinematics based on the new physical concepts of the relativity of space and time.<ref>Holton (1988), 196–206</ref><ref>Hentschel (1990), 3–13{{full citation needed|date=September 2019}}</ref><ref>Miller (1981), 216–217</ref><ref>Darrigol (2005), 15–18</ref><ref>Katzir (2005), 286–288</ref>

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 systems.

这些专著包括一个关于 Poincaré 的想法，这个想法后来成为数学“混沌理论”(特别是庞加莱始态复现定理)和动力系统的一般理论的基础。



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).

庞加莱为引力旋转流体的平衡图撰写了重要的天文学著作。他引入了分岔点的重要概念，证明了非椭球形平衡点的存在性及其稳定性。因为这个发现，庞加莱收到了英国皇家天文学会金质奖章。

While this is the view of most historians, a minority go much further, such as [[E. T. Whittaker]], who held that Poincaré and Lorentz were the true discoverers of relativity.<ref>Whittaker 1953, Secondary sources on relativity</ref>



===Algebra and number theory===

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。

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>]]

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

积分曲线的奇点" >



File: Phase Portrait Sadle.svg | Saddle

文件: Phase Portrait Sadle.svg | Saddle

===Topology===

File: Phase Portrait Stable Focus.svg | Focus

文件: 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}}

File: Phase portrait center.svg | Center

文件: Phase portrait Center.svg | Center



File: Phase Portrait Stable Node.svg | Node

文件: Phase Portrait Stable Node.svg | Node

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 >



===Astronomy and celestial mechanics===

[[File:N-body problem (3).gif|frame|left|150px | <center> Chaotic motion in three-body problem (computer simulation).</center>]]

Photographic portrait of H. Poincaré by Henri Manuel

庞加莱肖像摄影: Henri Manuel

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'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.

庞加莱的工作习惯被比作一只蜜蜂从一朵花飞到另一朵花。庞加莱对自己的思维方式很感兴趣; 他研究了自己的习惯，并于1908年在巴黎的普通心理学研究所就自己的观察发表了演讲。他把自己的思维方式与他如何做出几项发现联系起来。



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.

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.)

数学家 Darboux 声称他是非直觉的(一种直觉) ，认为这是由事实证明，他工作如此频繁的视觉表示。他不在乎严谨和不喜欢逻辑。(尽管有这样的观点，但雅克·阿达马写道，庞加莱的研究证明了非凡的清晰度，庞加莱本人写道，他 < ! -- todo: 添加庞加莱关于严格性的观点，见 http://www.forgottenbooks. org/readbook/american journal of mathematics 1890 v121000084889 # 233ー每一次我可以绝对严格 -- 相信逻辑不是一种发明方式，而是一种构造思想的方式，逻辑限制了思想。)

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>



===Differential equations and mathematical physics===

Poincaré's mental organisation was not only interesting to Poincaré himself but also to Édouard Toulouse, a psychologist of the Psychology Laboratory of the School of Higher Studies in Paris. Toulouse wrote a book entitled Henri Poincaré (1910). In it, he discussed Poincaré's regular schedule:

庞加莱的心理组织不仅引起了庞加莱本人的兴趣，也引起了巴黎高等研究学院心理学实验室的心理学家爱德华 · 图卢兹的兴趣。图卢兹写了一本书，名为《亨利 · 庞加莱》(Henri poincaré，1910)。在书中，他讨论了庞加莱的日程安排:

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).<ref>French: "Mémoire sur les courbes définies par une équation différentielle"</ref> 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.<ref>{{cite book|editor1-last=Kolmogorov|editor1-first = A.N.|editor2-first = A.P.|editor2-last= Yushkevich|title = Mathematics of the 19th century |volume= 3| pages = 162–174, 283|isbn= 978-3764358457|date = 24 March 1998}}</ref>



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

File: Phase Portrait Sadle.svg | Saddle

File: Phase Portrait Stable Focus.svg | Focus

File: Phase portrait center.svg | Center

These abilities were offset to some extent by his shortcomings:

这些能力在一定程度上被他的缺点所抵消:

File: Phase Portrait Stable Node.svg | Node

</gallery>



==Character==

[[File:Henri Poincaré by H Manuel.jpg|thumb|right|Photographic portrait of H. Poincaré by Henri Manuel]]

In addition, Toulouse stated that most mathematicians worked from principles already established while Poincaré started from basic principles each time (O'Connor et al., 2002).

此外，图卢兹说，大多数数学家从已经建立的原则开始工作，而庞加莱每次都从基本原则开始(奥康纳等人，2002年)。

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.



His method of thinking is well summarised as:

他的思维方式可以很好地概括为:

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.<ref>{{cite book |title=Encounter|volume = 12|author= Congress for Cultural Freedom|url=https://books.google.com/books?id=4-QLAQAAIAAJ&q=Poincaré+disliked+logic|year=1959|publisher=Martin Secker & Warburg.}}</ref> (Despite this opinion, [[Jacques Hadamard]] wrote that Poincaré's research demonstrated marvelous clarity<ref>J. Hadamard. L'oeuvre de H. Poincaré. Acta Mathematica, 38 (1921), p. 208</ref> 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.)



===Toulouse's characterisation===

Poincaré's mental organisation was not only interesting to Poincaré himself but also to Édouard Toulouse, a psychologist of the Psychology Laboratory of the School of Higher Studies in Paris. Toulouse wrote a book entitled ''Henri Poincaré'' (1910).<ref>[http://name.umdl.umich.edu/AAS9989.0001.001 Toulouse, Édouard, 1910. ''Henri Poincaré'', E. Flammarion, Paris]</ref><ref name="google">{{cite book|title=Henri Poincare|author=Toulouse, E.|date=2013|publisher=MPublishing|isbn=9781418165062|url=https://books.google.com/books?id=mpjWPQAACAAJ|accessdate=10 October 2014}}</ref> In it, he discussed Poincaré's regular schedule:

* He worked during the same times each day in short periods of time. He undertook mathematical research for four hours a day, between 10&nbsp;a.m. and noon then again from 5&nbsp;p.m. to 7&nbsp;p.m.. He would read articles in journals later in the evening.

Poincaré was dismayed by Georg Cantor's theory of transfinite numbers, and referred to it as a "disease" from which mathematics would eventually be cured.

庞加莱对康托的超限数理论感到沮丧，并称其为一种“疾病” ，数学最终将从中得到治愈。

* His normal work habit was to solve a problem completely in his head, then commit the completed problem to paper.

Poincaré said, "There is no actual infinite; the Cantorians have forgotten this, and that is why they have fallen into contradiction."

庞加莱说: “没有真正的无限，坎特利亚人已经忘记了这一点，这就是他们陷入矛盾的原因。”

* He was ambidextrous and nearsighted.

* His ability to visualise what he heard proved particularly useful when he attended lectures, since his eyesight was so poor that he could not see properly what the lecturer wrote on the blackboard.



Awards

奖项

These abilities were offset to some extent by his shortcomings:

* He was physically clumsy and artistically inept.

* He was always in a rush and disliked going back for changes or corrections.

* He never spent a long time on a problem since he believed that the subconscious would continue working on the problem while he consciously worked on another problem.



In addition, Toulouse stated that most mathematicians worked from principles already established while Poincaré started from basic principles each time (O'Connor et al., 2002).



His method of thinking is well summarised as:



{{quote|text=''Habitué à négliger les détails et à ne regarder que les cimes, il passait de l'une à l'autre avec une promptitude surprenante et les faits qu'il découvrait se groupant d'eux-mêmes autour de leur centre étaient instantanément et automatiquement classés dans sa mémoire.'' (Accustomed to neglecting details and to looking only at mountain tops, he went from one peak to another with surprising rapidity, and the facts he discovered, clustering around their center, were instantly and automatically pigeonholed in his memory.)|sign=Belliver (1956)}}



Named after him

以他的名字命名

===Attitude towards transfinite numbers===

Poincaré was dismayed by [[Georg Cantor]]'s theory of [[transfinite number]]s, and referred to it as a "disease" from which mathematics would eventually be cured.<ref name="daub266">Dauben 1979, p. 266.</ref>

Poincaré said, "There is no actual infinite; the Cantorians have forgotten this, and that is why they have fallen into contradiction."<ref>{{citation

|title=From Frege to Gödel: a source book in mathematical logic, 1879–1931

|first1=Jean

|last1=Van Heijenoort

|publisher=Harvard University Press

|year=1967

|isbn=978-0-674-32449-7

Henri Poincaré did not receive the Nobel Prize in Physics, but he had influential advocates like Henri Becquerel or committee member Gösta Mittag-Leffler. The nomination archive reveals that Poincaré received a total of 51 nominations between 1904 and 1912, the year of his death. Of the 58 nominations for the 1910 Nobel Prize, 34 named Poincaré. In Poincaré's case, several of those who nominated him pointed out that the greatest problem was to name a specific discovery, invention, or technique. Poincaré believed that Newton's first law was not empirical but is a conventional framework assumption for mechanics (Gargani, 2012). He also believed that the geometry of physical space is conventional. He considered examples in which either the geometry of the physical fields or gradients of temperature can be changed, either describing a space as non-Euclidean measured by rigid rulers, or as a Euclidean space where the rulers are expanded or shrunk by a variable heat distribution. However, Poincaré thought that we were so accustomed to Euclidean geometry that we would prefer to change the physical laws to save Euclidean geometry rather than shift to a non-Euclidean physical geometry.

没有获得诺贝尔物理学奖，但是他有一些有影响力的拥护者，比如 Henri Becquerel 或者委员会成员哥斯塔·米塔-列夫勒。提名档案显示，庞加莱在1904年至1912年间共获得51项提名。在1910年诺贝尔奖的58项提名中，有34项被提名为庞加莱。在 Poincaré 的案例中，一些提名他的人指出，最大的问题是命名一个具体的发现、发明或技术。庞加莱认为牛顿第一定律不是经验的，而是力学的常规框架假设(Gargani，2012)。他还认为物理空间的几何学是传统的。他考虑了一些例子，在这些例子中，物理场的几何形状或温度梯度可以改变，或者将一个空间描述为由刚性直尺测量的非欧几里德空间，或者将其描述为一个欧几里德空间，在这个空间中，直尺由变化的热分布而膨胀或收缩。然而，庞加莱认为我们已经习惯了欧几里得几何，我们宁愿改变物理定律来拯救欧几里得几何，而不是转向非欧几里德物理几何。

|page=190

|url=https://books.google.com/books?id=v4tBTBlU05sC&pg=PA190}}, [https://books.google.com/books?id=v4tBTBlU05sC&pg=PA190 p 190]

</ref>

Poincaré's famous lectures before the Société de Psychologie in Paris (published as Science and Hypothesis, The Value of Science, and Science and Method) were cited by Jacques Hadamard as the source for the idea that creativity and invention consist of two mental stages, first random combinations of possible solutions to a problem, followed by a critical evaluation.

庞加莱在巴黎 Société de Psychologie 之前的著名演讲(出版为《科学与假说》、《科学的价值》和《科学与方法》)被雅克·阿达马引用为创造力和发明由两个心理阶段组成，第一阶段是对问题可能解决方案的随机组合，随后是批判性评价。



==Honours==

Although he most often spoke of a deterministic universe, Poincaré said that the subconscious generation of new possibilities involves chance.

尽管庞加莱最常提到的是确定性的宇宙，但他说潜意识中产生的新的可能性包含着机会。

'''Awards'''

<blockquote>

< 封锁报价 >

*Oscar II, King of Sweden's mathematical competition (1887)

It is certain that the combinations which present themselves to the mind in a kind of sudden illumination after a somewhat prolonged period of unconscious work are generally useful and fruitful combinations... all the combinations are formed as a result of the automatic action of the subliminal ego, but those only which are interesting find their way into the field of consciousness... A few only are harmonious, and consequently at once useful and beautiful, and they will be capable of affecting the geometrician's special sensibility I have been speaking of; which, once aroused, will direct our attention upon them, and will thus give them the opportunity of becoming conscious... In the subliminal ego, on the contrary, there reigns what I would call liberty, if one could give this name to the mere absence of discipline and to disorder born of chance.

可以肯定的是，在经过一段时间的无意识工作之后，以一种突然的启发出现在头脑中的组合，通常是有用的和富有成效的组合... ... 所有的组合都是潜意识自动作用的结果，但是只有那些有趣的组合才能进入意识领域... ... 只有少数组合是和谐的，因此是有用的和美丽的，它们将能够影响我所说的几何学家的特殊感受力; 一旦唤起我们的注意力，就会引导我们对它们的注意力，从而给它们成为意识的机会... ..。恰恰相反，在潜意识中，如果一个人可以把缺乏纪律和偶然产生的混乱称为自由，那么我将称之为自由。

*Foreign member of the [[Royal Netherlands Academy of Arts and Sciences]] (1897)<ref>{{cite web |url=http://www.dwc.knaw.nl/biografie/pmknaw/?pagetype=authorDetail&aId=PE00002358 |title=Jules Henri Poincaré (1854–1912) |publisher=Royal Netherlands Academy of Arts and Sciences |date= |accessdate=4 August 2015 |archive-url=https://web.archive.org/web/20150905152142/http://www.dwc.knaw.nl/biografie/pmknaw/?pagetype=authorDetail&aId=PE00002358 |archive-date=5 September 2015 |url-status=dead }}</ref>

</blockquote>

</blockquote >

*[[American Philosophical Society]] 1899

*[[Gold Medal of the Royal Astronomical Society]] of London (1900)

Poincaré's two stages—random combinations followed by selection—became the basis for Daniel Dennett's two-stage model of free will.

庞加莱的两个阶段——随机组合后是选择——成为丹尼尔 · 丹尼特两阶段自由意志模型的基础。

*[[Bolyai Prize]] in 1905

*[[Matteucci Medal]] 1905

*[[French Academy of Sciences]] 1906

*[[Académie française]] 1909

*[[Bruce Medal]] (1911)

Popular writings on the philosophy of science:

关于科学哲学的通俗著作:



'''Named after him'''

|author=Poincaré, Henri

| author = poincaré，Henri

*[[Institut Henri Poincaré]] (mathematics and theoretical physics center)

|year=1902–1908

| year = 1902-1908

*[[Poincaré Prize]] (Mathematical Physics International Prize)

|title=The Foundations of Science

科学的基础

*[[Annales Henri Poincaré]] (Scientific Journal)

|place=New York

地点: 纽约

*Poincaré Seminar (nicknamed "[[Bourbaphy]]")

|publisher=Science Press

科学出版社

*The crater [[Poincaré (crater)|Poincaré]] on the Moon

|url=https://archive.org/details/foundationsscie01poingoog}}; reprinted in 1921; This book includes the English translations of Science and Hypothesis (1902), The Value of Science (1905), Science and Method (1908).

这本书包括《科学与假说》(Science and Hypothesis)(1902)、《科学的价值》(The Value of Science)(1905)、《科学与方法》(Science and Method)(1908)的英译 https://archive.org/details/foundationsscie01poingoog。

*[[Asteroid]] [[2021 Poincaré]]

*[[List of things named after Henri Poincaré]]



Henri Poincaré did not receive the [[Nobel Prize in Physics]], but he had influential advocates like [[Henri Becquerel]] or committee member [[Gösta Mittag-Leffler]].<ref name="gray-biography">{{cite book|last1=Gray|first1=Jeremy|title=Henri Poincaré: A Scientific Biography|date=2013|publisher=Princeton University Press|pages=194–196|chapter=The Campaign for Poincaré}}</ref><ref>{{cite book|last1=Crawford|first1=Elizabeth|title=The Beginnings of the Nobel Institution: The Science Prizes, 1901–1915|date=25 November 1987|publisher=Cambridge University Press|pages=141–142}}</ref> The nomination archive reveals that Poincaré received a total of 51 nominations between 1904 and 1912, the year of his death.<ref name="nomination database">{{cite web|title=Nomination database|url=https://www.nobelprize.org/nomination/archive/list.php|website=Nobelprize.org|publisher=Nobel Media AB|accessdate=24 September 2015}}</ref> Of the 58 nominations for the 1910 Nobel Prize, 34 named Poincaré.<ref name="nomination database"/> Nominators included Nobel laureates [[Hendrik Lorentz]] and [[Pieter Zeeman]] (both of 1902), [[Marie Curie]] (of 1903), [[Albert Michelson]] (of 1907), [[Gabriel Lippmann]] (of 1908) and [[Guglielmo Marconi]] (of 1909).<ref name="nomination database"/>



The fact that renowned theoretical physicists like Poincaré, Boltzmann or Gibbs were not awarded the Nobel Prize is seen as evidence that the Nobel committee had more regard for experimentation than theory.<ref>{{cite journal|last1=Crawford |first1= Elizabeth |title=Nobel: Always the Winners, Never the Losers|journal=[[Science (journal)|Science]]|date=13 November 1998|volume=282|issue=5392|pages=1256–1257|doi=10.1126/science.282.5392.1256|bibcode = 1998Sci...282.1256C |s2cid= 153619456 }}{{dead link|date=July 2016}}</ref><ref>{{cite journal|last1=Nastasi|first1=Pietro|title=A Nobel Prize for Poincaré? |journal=Lettera Matematica|date=16 May 2013|volume=1|issue=1–2|pages=79–82|doi=10.1007/s40329-013-0005-1 |url= |accessdate=|doi-access=free}}</ref> In Poincaré's case, several of those who nominated him pointed out that the greatest problem was to name a specific discovery, invention, or technique.<ref name="gray-biography"/>



==Philosophy==

On algebraic topology:

关于代数拓扑:

Poincaré had philosophical views opposite to those of [[Bertrand Russell]] and [[Gottlob Frege]], who believed that mathematics was a branch of [[logic]]. Poincaré strongly disagreed, claiming that [[intuition (knowledge)|intuition]] was the life of mathematics. Poincaré gives an interesting point of view in his book ''[[Science and Hypothesis]]'':



| url=http://www.maths.ed.ac.uk/~aar/papers/poincare2009.pdf}}. The first systematic study of topology.

Http://www.maths.ed.ac.uk/~aar/papers/poincare2009.pdf.第一个系统的拓扑学研究。

{{quote|text=For a superficial observer, scientific truth is beyond the possibility of doubt; the logic of science is infallible, and if the scientists are sometimes mistaken, this is only from their mistaking its rule.}}



On celestial mechanics:

关于天体力学:

Poincaré believed that [[arithmetic]] is [[Analytic/synthetic distinction|synthetic]]. He argued that [[Peano's axioms]] cannot be proven non-circularly with the principle of induction (Murzi, 1998), therefore concluding that arithmetic is ''[[A priori and a posteriori|a priori]]'' synthetic and not analytic. Poincaré then went on to say that mathematics cannot be deduced from logic since it is not analytic. His views were similar to those of [[Immanuel Kant]] (Kolak, 2001, Folina 1992). He strongly opposed Cantorian [[set theory]], objecting to its use of [[Impredicativity|impredicative]] definitions{{Citation needed|date=March 2018}}.



However, Poincaré did not share Kantian views in all branches of philosophy and mathematics. For example, in geometry, Poincaré believed that the structure of [[Non-Euclidean geometry|non-Euclidean space]] can be known analytically. Poincaré held that convention plays an important role in physics. His view (and some later, more extreme versions of it) came to be known as "[[conventionalism]]".<ref>Yemima Ben-Menahem, ''Conventionalism: From Poincare to Quine'', Cambridge University Press, 2006, p. 39.</ref> Poincaré believed that [[Newton's first law]] was not empirical but is a conventional framework assumption for mechanics (Gargani, 2012).<ref>{{Citation|author=Gargani Julien|title=Poincaré, le hasard et l'étude des systèmes complexes|publisher=L'Harmattan|year=2012|page=124|url=http://www.editions-harmattan.fr/index.asp?navig=catalogue&obj=livre&no=38754|access-date=5 June 2015|archive-url=https://web.archive.org/web/20160304140554/http://www.editions-harmattan.fr/index.asp?navig=catalogue&obj=livre&no=38754|archive-date=4 March 2016|url-status=dead}}</ref> He also believed that the geometry of physical space is conventional. He considered examples in which either the geometry of the physical fields or gradients of temperature can be changed, either describing a space as non-Euclidean measured by rigid rulers, or as a Euclidean space where the rulers are expanded or shrunk by a variable heat distribution. However, Poincaré thought that we were so accustomed to [[Euclidean geometry]] that we would prefer to change the physical laws to save Euclidean geometry rather than shift to a non-Euclidean physical geometry.<ref>{{Citation|title=Science and Hypothesis|first1=Henri |last1=Poincaré |publisher=Cosimo, Inc. Press|year=2007|isbn=978-1-60206-505-5 |page=50

|url=https://books.google.com/books?id=2QXqHaVbkgoC&pg=PA50}}</ref>



On the philosophy of mathematics:

关于数学哲学:

===Free will===

Poincaré's famous lectures before the Société de Psychologie in Paris (published as ''[[Science and Hypothesis]]'', ''[[The Value of Science]]'', and ''Science and Method'') were cited by [[Jacques Hadamard]] as the source for the idea that creativity and invention consist of two mental stages, first random combinations of possible solutions to a problem, followed by a critical evaluation.<ref>Hadamard, Jacques. ''An Essay on the Psychology of Invention in the Mathematical Field''. Princeton Univ Press (1945)</ref>



Although he most often spoke of a deterministic universe, Poincaré said that the subconscious generation of new possibilities involves [[Randomness|chance]].

<blockquote>

It is certain that the combinations which present themselves to the mind in a kind of sudden illumination after a somewhat prolonged period of unconscious work are generally useful and fruitful combinations... all the combinations are formed as a result of the automatic action of the subliminal ego, but those only which are interesting find their way into the field of consciousness... A few only are harmonious, and consequently at once useful and beautiful, and they will be capable of affecting the geometrician's special sensibility I have been speaking of; which, once aroused, will direct our attention upon them, and will thus give them the opportunity of becoming conscious... In the subliminal ego, on the contrary, there reigns what I would call liberty, if one could give this name to the mere absence of discipline and to disorder born of chance.<ref>{{cite book|title =Science and Method|chapter= 3: Mathematical Creation|date= 1914|chapter-url = https://ebooks.adelaide.edu.au/p/poincare/henri/science-and-method/book1.3.html|first = Henri|last =Poincaré }}</ref>

</blockquote>



Poincaré's two stages—random combinations followed by selection—became the basis for [[Daniel Dennett]]'s two-stage model of free will.<ref>Dennett, Daniel C. 1978. Brainstorms: Philosophical Essays on Mind and Psychology. The MIT Press, p.293</ref>



==Bibliography==

Other:

其他:

===Poincaré's writings in English translation===



Popular writings on the [[philosophy of science]]:

*{{Citation

|author=Poincaré, Henri

Exhaustive bibliography of English translations:

详尽的英语翻译书目:

|year=1902–1908

|title=The Foundations of Science

|place=New York

|publisher=Science Press

|url=https://archive.org/details/foundationsscie01poingoog}}; reprinted in 1921; This book includes the English translations of Science and Hypothesis (1902), The Value of Science (1905), Science and Method (1908).

* 1904. ''Science and Hypothesis,'' The Walter Scott Publishing Co.

* 1913. "The New Mechanics," The Monist, Vol. XXIII.

* 1913. "The Relativity of Space," The Monist, Vol. XXIII.

* 1913. {{Citation | title=Last Essays. |place=New York |publisher=Dover reprint, 1963 | url=https://archive.org/details/mathematicsandsc001861mbp}}

* 1956. ''Chance.'' In James R. Newman, ed., The World of Mathematics (4 Vols).

* 1958. ''The Value of Science,'' New York: Dover.



On [[algebraic topology]]:

* 1895. {{Citation |title=Analysis Situs

| url=http://www.maths.ed.ac.uk/~aar/papers/poincare2009.pdf}}. The first systematic study of [[topology]].



On [[celestial mechanics]]:

* 1892–99. ''New Methods of Celestial Mechanics'', 3 vols. English trans., 1967. {{isbn|1-56396-117-2}}.

* 1905. "The Capture Hypothesis of J. J. See," The Monist, Vol. XV.

* 1905–10. ''Lessons of Celestial Mechanics''.



On the [[philosophy of mathematics]]:

* Ewald, William B., ed., 1996. ''From Kant to Hilbert: A Source Book in the Foundations of Mathematics'', 2 vols. Oxford Univ. Press. Contains the following works by Poincaré:

** 1894, "On the Nature of Mathematical Reasoning," 972–81.

** 1898, "On the Foundations of Geometry," 982–1011.

** 1900, "Intuition and Logic in Mathematics," 1012–20.

** 1905–06, "Mathematics and Logic, I–III," 1021–70.

** 1910, "On Transfinite Numbers," 1071–74.

* 1905. "The Principles of Mathematical Physics," The Monist, Vol. XV.

* 1910. "The Future of Mathematics," The Monist, Vol. XX.

* 1910. "Mathematical Creation," The Monist, Vol. XX.



Other:

* 1904. ''Maxwell's Theory and Wireless Telegraphy,'' New York, McGraw Publishing Company.

* 1905. "The New Logics," The Monist, Vol. XV.

* 1905. "The Latest Efforts of the Logisticians," The Monist, Vol. XV.



Exhaustive bibliography of English translations:

{{Columns-list|colwidth=30em|

{{ Columns-list | colwidth = 30em |

* 1892–2017. {{Citation |title=Henri Poincaré Papers |url=http://henripoincarepapers.univ-nantes.fr/bibliohp/index.php?a=on&lang=en&action=Chercher }}{{Dead link|date=May 2020 |bot=InternetArchiveBot |fix-attempted=yes }}.



==See also==

===Concepts===

* [[Poincaré complex]] – an abstraction of the singular chain complex of a closed, orientable manifold

* [[Poincaré duality]]

* [[Poincaré disk model]]

* [[Poincaré group]]

}}

}}

* [[Poincaré half-plane model]]

* [[Homology sphere#Poincaré homology sphere|Poincaré homology sphere]]

* [[Poincaré inequality]]

* [[Poincaré map]]

* [[Poincaré residue]]

* [[Poincaré series (modular form)]]

* [[Poincaré space]]

* [[Poincaré metric]]

* [[Poincaré plot]]

* [[Hilbert–Poincaré series|Poincaré series]]

* [[Poincaré sphere (optics)|Poincaré sphere]]

* [[Poincaré–Lelong equation]]

* [[Poincaré–Lindstedt method]]

* [[Poincaré–Lindstedt perturbation theory]]

* [[Poincaré–Steklov operator]]

* [[Reflecting Function]]



===Theorems===

|title=Henri Poincaré. A Life in the Service of Science

|title=Henri Poincaré.为科学服务的一生

* [[Poincaré's recurrence theorem]]: certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state.

|author=Jean Mawhin |journal=Notices of the AMS

作者: Jean Mawhin | journal = AMS 公告

* [[Poincaré–Bendixson theorem]]: a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere.

|date=October 2005 |volume=52 |issue=9 |pages=1036–1044 }}

| date = October 2005 | volume = 52 | issue = 9 | pages = 1036-1044}

* [[Poincaré–Hopf theorem]]: a generalization of the hairy-ball theorem, which states that there is no smooth vector field on a sphere having no sources or sinks.

* [[Poincaré–Lefschetz duality theorem]]: a version of Poincaré duality in geometric topology, applying to a manifold with boundary

* [[Poincaré separation theorem]]: gives the upper and lower bounds of eigenvalues of a real symmetric matrix B'AB that can be considered as the orthogonal projection of a larger real symmetric matrix A onto a linear subspace spanned by the columns of B.

* [[Poincaré–Birkhoff theorem]]: every area-preserving, orientation-preserving homeomorphism of an annulus that rotates the two boundaries in opposite directions has at least two fixed points.

* [[Poincaré–Birkhoff–Witt theorem]]: an explicit description of the universal enveloping algebra of a Lie algebra.

* [[Poincaré conjecture]] (now a theorem): Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

* [[Poincaré–Miranda theorem]]: a generalization of the [[intermediate value theorem]] to ''n'' dimensions.



=== Other ===

{{Columns-list|colwidth=30em|

*[[French epistemology]]

*[[History of special relativity]]

*[[List of things named after Henri Poincaré]]

*[[Institut Henri Poincaré]], Paris

*[[Brouwer fixed-point theorem]]

*[[Relativity priority dispute]]

*[[Epistemic structural realism]]<ref>[http://plato.stanford.edu/entries/structural-realism/#Rel "Structural Realism"]: entry by James Ladyman in the ''[[Stanford Encyclopedia of Philosophy]]''</ref>

}}



==References==



===Footnotes===

{{Reflist}}



===Sources===

* [[Eric Temple Bell|Bell, Eric Temple]], 1986. ''Men of Mathematics'' (reissue edition). Touchstone Books. {{isbn|0-671-62818-6}}.

* Belliver, André, 1956. ''Henri Poincaré ou la vocation souveraine''. Paris: Gallimard.

*[[Peter L. Bernstein|Bernstein, Peter L]], 1996. "Against the Gods: A Remarkable Story of Risk". (p.&nbsp;199–200). John Wiley & Sons.

* [[Carl Benjamin Boyer|Boyer, B. Carl]], 1968. ''A History of Mathematics: Henri Poincaré'', John Wiley & Sons.

* [[Ivor Grattan-Guinness|Grattan-Guinness, Ivor]], 2000. ''The Search for Mathematical Roots 1870–1940.'' Princeton Uni. Press.

* {{Citation|last=Dauben|given=Joseph|authorlink=Joseph Dauben|origyear=1993|year=2004|chapter=Georg Cantor and the Battle for Transfinite Set Theory|chapter-url=http://www.acmsonline.org/journal/2004/Dauben-Cantor.pdf|title=Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, CA)|pages=1–22|url-status=dead|archiveurl=https://web.archive.org/web/20100713115605/http://www.acmsonline.org/journal/2004/Dauben-Cantor.pdf|archivedate=13 July 2010}}. Internet version published in Journal of the ACMS 2004.

* Folina, Janet, 1992. ''Poincaré and the Philosophy of Mathematics.'' Macmillan, New York.

* [[Jeremy Gray|Gray, Jeremy]], 1986. ''Linear differential equations and group theory from Riemann to Poincaré'', Birkhauser {{isbn|0-8176-3318-9}}

* Gray, Jeremy, 2013. ''Henri Poincaré: A scientific biography''. Princeton University Press {{isbn|978-0-691-15271-4}}

*{{Citation |url=http://www.ams.org/notices/200509/comm-mawhin.pdf

|title=Henri Poincaré. A Life in the Service of Science

|author=Jean Mawhin |journal=Notices of the AMS

|date=October 2005 |volume=52 |issue=9 |pages=1036–1044 }}

* Kolak, Daniel, 2001. ''Lovers of Wisdom'', 2nd ed. Wadsworth.

* Gargani, Julien, 2012. ''Poincaré, le hasard et l'étude des systèmes complexes'', L'Harmattan.

* Murzi, 1998. "Henri Poincaré".

* O'Connor, J. John, and Robertson, F. Edmund, 2002, "Jules Henri Poincaré". University of St. Andrews, Scotland.

* [[Ivars Peterson|Peterson, Ivars]], 1995. ''Newton's Clock: Chaos in the Solar System'' (reissue edition). W H Freeman & Co. {{isbn|0-7167-2724-2}}.

* Sageret, Jules, 1911. ''Henri Poincaré''. Paris: Mercure de France.

* Toulouse, E.,1910. ''Henri Poincaré''.—(Source biography in French) at University of Michigan Historic Math Collection.

* {{cite book |title=Mathematics and Its History |edition=3rd, illustrated |first1=John |last1=Stillwell |publisher= Springer Science & Business Media |year=2010 |isbn=978-1-4419-6052-8 |url=https://books.google.com/books?id=V7mxZqjs5yUC |ref=harv}}

* [[F. Verhulst|Verhulst, Ferdinand]], 2012 ''Henri Poincaré. Impatient Genius''. N.Y.: Springer.

* ''Henri Poincaré, l'œuvre scientifique, l'œuvre philosophique'', by Vito Volterra, Jacques Hadamard, Paul Langevin and Pierre Boutroux, Felix Alcan, 1914.

** ''Henri Poincaré, l'œuvre mathématique'', by [[Vito Volterra]].

** ''Henri Poincaré, le problème des trois corps'', by [[Jacques Hadamard]].

** ''Henri Poincaré, le physicien'', by [[Paul Langevin]].

** ''Henri Poincaré, l'œuvre philosophique'', by [[Pierre Boutroux]].

* {{PlanetMath attribution|id=3793|title=Jules Henri Poincaré}}



==Further reading==

===Secondary sources to work on relativity===

* {{Citation | author=Cuvaj, Camillo | year=1969 | title= Henri Poincaré's Mathematical Contributions to Relativity and the Poincaré Stresses | journal=American Journal of Physics |pages=1102–1113 |volume=36 | issue=12|doi=10.1119/1.1974373|bibcode = 1968AmJPh..36.1102C }}

* {{Citation|author=Darrigol, O. |title=Henri Poincaré's criticism of Fin De Siècle electrodynamics |year=1995 |journal=Studies in History and Philosophy of Science |volume=26|issue=1|pages=1–44|doi=10.1016/1355-2198(95)00003-C|bibcode=1995SHPMP..26....1D}}

* {{Citation | author=Darrigol, O. | year=2000 | title=Electrodynamics from Ampére to Einstein | place=Oxford | publisher=Clarendon Press | isbn=978-0-19-850594-5 | url-access=registration | url=https://archive.org/details/electrodynamicsf0000darr }}

* {{Citation|author=Darrigol, O. |title=The Mystery of the Einstein–Poincaré Connection| pages=614–626|doi=10.1086/430652|pmid=16011297 |year=2004 |journal=Isis|volume=95| issue=4|s2cid=26997100}}

* {{Citation|author=Darrigol, O. |title=The Genesis of the theory of relativity |year=2005 |journal=Séminaire Poincaré|volume=1|pages=1–22|url=http://www.bourbaphy.fr/darrigol2.pdf|doi=10.1007/3-7643-7436-5_1|isbn=978-3-7643-7435-8 |bibcode=2006eins.book....1D }}

* {{Citation | author=Galison, P. | year=2003 | title= Einstein's Clocks, Poincaré's Maps: Empires of Time | place=New York |publisher=W.W. Norton|isbn=978-0-393-32604-8}}

* {{Citation|author=Giannetto, E. |title=The Rise of Special Relativity: Henri Poincaré's Works Before Einstein |year=1998 |journal=Atti del XVIII Congresso di Storia della Fisica e dell'astronomia |pages=171–207}}

* {{Citation | author=Giedymin, J. | year=1982 | title= Science and Convention: Essays on Henri Poincaré's Philosophy of Science and the Conventionalist Tradition | place=Oxford |publisher=Pergamon Press|isbn=978-0-08-025790-7| author-link=Jerzy Giedymin }}

* {{Citation | author=Goldberg, S. | year=1967 | title= Henri Poincaré and Einstein's Theory of Relativity | journal=American Journal of Physics |pages=934–944 |volume=35 | issue=10|doi=10.1119/1.1973643|bibcode = 1967AmJPh..35..934G }}

* {{Citation | author=Goldberg, S. | year=1970 | title= Poincaré's silence and Einstein's relativity | journal=British Journal for the History of Science |pages=73–84 |volume=5 | doi=10.1017/S0007087400010633}}

*{{Citation | author=Holton, G. | origyear=1973| year=1988 | chapter=Poincaré and Relativity| title= Thematic Origins of Scientific Thought: Kepler to Einstein | publisher=Harvard University Press|isbn=978-0-674-87747-4}}

* {{Citation | author=Katzir, S. | year=2005 | journal=Phys. Perspect. | title= Poincaré's Relativistic Physics: Its Origins and Nature |pages= 268–292 |volume=7 | doi=10.1007/s00016-004-0234-y | issue=3 |bibcode = 2005PhP.....7..268K | s2cid=14751280 }}

* {{Citation|author=Keswani, G.H., Kilmister, C.W. |year=1983 |journal=Br. J. Philos. Sci. |title=Intimations of Relativity: Relativity Before Einstein |pages=343–354 |volume=34 |doi=10.1093/bjps/34.4.343 |issue=4 |url=http://osiris.sunderland.ac.uk/webedit/allweb/news/Philosophy_of_Science/PIRT2002/Intimations%20of%20Relativity.doc |url-status=dead |archiveurl=https://web.archive.org/web/20090326084436/http://osiris.sunderland.ac.uk/webedit/allweb/news/Philosophy_of_Science/PIRT2002/Intimations%20of%20Relativity.doc |archivedate=26 March 2009}}

* {{Citation | author=Keswani, G.H. | year=1965| journal=Br. J. Philos. Sci. | title= Origin and Concept of Relativity, Part I |volume=15| issue=60|pages=286–306 |doi=10.1093/bjps/XV.60.286}}

* {{Citation | author=Keswani, G.H. | year=1965 | journal=Br. J. Philos. Sci. | title= Origin and Concept of Relativity, Part II|volume=16| pages=19–32| issue=61| doi=10.1093/bjps/XVI.61.19}}

* {{Citation | author=Keswani, G.H. | year=1966 | journal=Br. J. Philos. Sci. | title= Origin and Concept of Relativity, Part III |volume=16|issue=64| pages=273–294| doi=10.1093/bjps/XVI.64.273 }}

* {{Citation | author=Kragh, H. | year=1999 | title= Quantum Generations: A History of Physics in the Twentieth Century |publisher= Princeton University Press|isbn=978-0-691-09552-3}}

* {{Citation | author=Langevin, P. | year=1913 | journal=Revue de Métaphysique et de Morale | title= L'œuvre d'Henri Poincaré: le physicien |page= 703 |volume=21|url=http://gallica.bnf.fr/ark:/12148/bpt6k111418/f93.chemindefer}}

* {{Citation | author=Macrossan, M. N. | year=1986 | journal=Br. J. Philos. Sci. | title=A Note on Relativity Before Einstein | pages=232–234 | volume=37 | issue=2 | url=http://espace.library.uq.edu.au/view.php?pid=UQ:9560 | doi=10.1093/bjps/37.2.232 | citeseerx=10.1.1.679.5898 | access-date=27 March 2007 | archive-url=https://web.archive.org/web/20131029203003/http://espace.library.uq.edu.au/view.php?pid=UQ:9560 | archive-date=29 October 2013 | url-status=dead }}

*{{Citation|author=Miller, A.I. |title=A study of Henri Poincaré's "Sur la Dynamique de l'Electron |year=1973 |journal=Arch. Hist. Exact Sci.|volume=10|pages=207–328|doi=10.1007/BF00412332|issue=3–5|s2cid=189790975 }}

* {{Citation | author=Miller, A.I. | year=1981 | title=Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911) | place=Reading | publisher=Addison–Wesley | isbn=978-0-201-04679-3 | url-access=registration | url=https://archive.org/details/alberteinsteinss0000mill }}

*{{Citation| author=Miller, A.I. |contribution= Why did Poincaré not formulate special relativity in 1905? |year=1996 |editor1=Jean-Louis Greffe |editor2=Gerhard Heinzmann |editor3=Kuno Lorenz | title=Henri Poincaré : science et philosophie| pages=69–100|place=Berlin}}

* {{Citation | author=Schwartz, H. M. | year=1971 | title= Poincaré's Rendiconti Paper on Relativity. Part I | journal=American Journal of Physics |pages=1287–1294 |volume=39 | issue=7|doi=10.1119/1.1976641|bibcode = 1971AmJPh..39.1287S }}

* {{Citation | author=Schwartz, H. M. | year=1972 | title= Poincaré's Rendiconti Paper on Relativity. Part II | journal=American Journal of Physics |pages=862–872 |volume=40 | issue=6| doi=10.1119/1.1986684|bibcode = 1972AmJPh..40..862S }}

* {{Citation | author=Schwartz, H. M. | year=1972 | title= Poincaré's Rendiconti Paper on Relativity. Part III | journal=American Journal of Physics |pages=1282–1287 |volume=40 | issue=9| doi=10.1119/1.1986815|bibcode = 1972AmJPh..40.1282S }}

* {{Citation | author=Scribner, C. | year=1964 | title= Henri Poincaré and the principle of relativity | journal=American Journal of Physics |pages=672–678 |volume=32 | issue=9| doi=10.1119/1.1970936|bibcode =1964AmJPh..32..672S }}

* {{Citation | author=Walter, S. | year=2005 | editor=Renn, J. | contribution= Henri Poincaré and the theory of relativity | title=Albert Einstein, Chief Engineer of the Universe: 100 Authors for Einstein |pages=162–165 | place=Berlin | publisher=Wiley-VCH|contribution-url=http://scottwalter.free.fr/papers/2005-100authors-poincare-einstein-walter.html}}

* {{Citation | author=Walter, S. | year=2007 | editor=Renn, J. | contribution= Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905–1910 | title=The Genesis of General Relativity |pages=193–252 |volume=3 |place=Berlin | publisher=Springer|contribution-url=http://scottwalter.free.fr/papers/2007-genesis-walter.html}}

* {{Citation | author=Whittaker, E.T.|authorlink=E. T. Whittaker | year=1953 | title= [[A History of the Theories of Aether and Electricity|A History of the Theories of Aether and Electricity: The Modern Theories 1900–1926]]| chapter= The Relativity Theory of Poincaré and Lorentz | place=London |publisher=Nelson}}

* {{Citation| author=Zahar, E. |year=2001 |title=Poincaré's Philosophy: From Conventionalism to Phenomenology |publisher=Open Court Pub Co|place=Chicago|isbn=978-0-8126-9435-2}}

Category:1854 births

类别: 出生人数1854人



Category:1912 deaths

分类: 1912年死亡人数

===Non-mainstream sources===

Category:19th-century French mathematicians

范畴: 19世纪法国数学家

* {{Citation | author=Leveugle, J. | year=2004 |title= La Relativité et Einstein, Planck, Hilbert—Histoire véridique de la Théorie de la Relativitén | publisher=L'Harmattan| place=Pars}}

Category:20th-century French philosophers

范畴: 20世纪法国哲学家

* {{Citation | author=Logunov, A.A. | year=2004 | title= Henri Poincaré and relativity theory | pages=<!-- --> | arxiv=physics/0408077 |bibcode = 2004physics...8077L |isbn=978-5-02-033964-4}}

Category:20th-century French mathematicians

范畴: 20世纪法国数学家



Category:Algebraic geometers

类别: 代数几何

==External links==

Category:Burials at Montparnasse Cemetery

类别: 蒙帕纳斯公墓的葬礼

{{commons|Henri Poincaré}}

Category:Chaos theorists

范畴: 混沌理论家

{{wikiquote}}

Category:Corps des mines

类别: 水雷部队

{{wikisource author}}

Category:Corresponding Members of the St Petersburg Academy of Sciences

类别: 圣彼得堡科学院通讯员

* {{Gutenberg author |id=Poincaré,+Henri | name=Henri Poincaré}}

Category:École Polytechnique alumni

类别: 巴黎综合理工学院校友

* {{Internet Archive author |sname=Henri Poincaré |sopt=w}}

Category:Foreign associates of the National Academy of Sciences

类别: 美国国家科学院的外国合伙人

* {{Librivox author |id=4281}}

Category:Foreign Members of the Royal Society

类别: 皇家学会的外国成员

*[[Internet Encyclopedia of Philosophy]]: "[http://www.utm.edu/research/iep/p/poincare.htm Henri Poincaré]"—by Mauro Murzi.

Category:French military personnel of the Franco-Prussian War

类别: 普法战争法国军事人员

*[[Internet Encyclopedia of Philosophy]]: "[http://www.iep.utm.edu/poi-math/ Poincaré’s Philosophy of Mathematics]"—by Janet Folina.

Category:French physicists

类别: 法国物理学家

* {{MathGenealogy |id=34227}}

Category:Geometers

类别: 几何学家

*[https://web.archive.org/web/20090930005045/https://www.informationphilosopher.com/solutions/scientists/poincare/ Henri Poincaré on Information Philosopher]

Category:Mathematical analysts

类别: 数学分析师

* {{MacTutor Biography|id=Poincare}}

Category:Members of the Académie Française

分类: 美国法兰西学术院协会会员

*[http://henripoincarepapers.univ-nantes.fr/chronos.php A timeline of Poincaré's life] University of Nantes (in French).

Category:Members of the Royal Netherlands Academy of Arts and Sciences

类别: 荷兰皇家艺术与科学学院成员

*[http://henripoincarepapers.univ-nantes.fr Henri Poincaré Papers] University of Nantes (in French).

Category:Mines ParisTech alumni

类别: Mines ParisTech alumni

*[https://web.archive.org/web/20060627062431/https://www.phys-astro.sonoma.edu/BruceMedalists/Poincare/index.html Bruce Medal page]

Category:Officers of the French Academy of Sciences

类别: 法国科学院官员

*Collins, Graham P., "[https://web.archive.org/web/20071017055831/http://www.sciam.com/print_version.cfm?articleID=0003848D-1C61-10C7-9C6183414B7F0000 Henri Poincaré, His Conjecture, Copacabana and Higher Dimensions]," ''[[Scientific American]]'', 9 June 2004.

Category:People from Nancy, France

类别: 来自法国南希的人

*BBC in Our Time, "[https://web.archive.org/web/20090424054425/http://www.bbc.co.uk/radio4/history/inourtime/inourtime.shtml Discussion of the Poincaré conjecture]," 2 November 2006, hosted by [[Melvynn Bragg]].

Category:Philosophers of science

范畴: 科学哲学家

*[https://web.archive.org/web/20070927190224/http://www.mathpages.com/home/kmath305/kmath305.htm Poincare Contemplates Copernicus] at MathPages

Category:Recipients of the Bruce Medal

类别: 布鲁斯奖章获得者

*[https://www.youtube.com/user/thedebtgeneration?feature=mhum#p/u/8/5pKrKdNclYs0 High Anxieties&nbsp;– The Mathematics of Chaos] (2008) BBC documentary directed by [[David Malone (independent filmmaker)|David Malone]] looking at the influence of Poincaré's discoveries on 20th Century mathematics.

Category:Recipients of the Gold Medal of the Royal Astronomical Society

类别: 英国皇家天文学会金质奖章奖学金获得者



Category:Relativity theorists

范畴: 相对论理论家

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Category:Thermodynamicists

类别: 热力学家

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Category:Fluid dynamicists

类别: 流体动力学家

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Category:Topologists

类别: 拓扑学家

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Category:University of Paris faculty

类别: 巴黎大学教员

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Category:French male writers

类别: 法国男性作家

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Category:Deaths from embolism

类别: 死于栓塞

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Category:Dynamical systems theorists

范畴: 动力系统理论家

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<small>This page was moved from [[wikipedia:en:Henri Poincaré]]. Its edit history can be viewed at [[庞加莱/edithistory]]</small></noinclude>

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