亨利·庞加莱 Jules Henri Poincaré

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{{简述}法国数学家、物理学家、工程师和科学哲学家}} 模板:More citations needed {{更多引文{日期=2017年4月}} 模板:使用dmy日期

{{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
(photograph published in 1913)

|caption = Henri Poincaré
(photograph published in 1913)

摄于1913年

|birth_date = (1854-模板:MONTHNUMBER-29)29 1854

|birth_date =

出生日期

|birth_place = 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 = 17 July 1912(1912-07-17) (aged 58)

|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

|nationality = French

| 国籍: 法国

|fields = Mathematics and physics

|fields = Mathematics and physics

| fields = 数学和物理

|workplaces = {{plainlist|

|workplaces =

• Corps des Mines

• Caen University

• La Sorbonne

• Bureau des Longitudes

|education = {{plainlist|

|education =

2009年10月11日

• Lycée Nancy (now 模板:Ill)

• École Polytechnique

• École des Mines

• University of Paris (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 =

Jules Henri Poincaré (模板:IPAc-en [US: stress final syllable], 模板:IPA-fr; 29 April 1854 – 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.

Jules Henri Poincaré ( [US: stress final syllable], ; 29 April 1854 – 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Jules Henri Poincaré 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” ，因为他在他有生之年在所有学科领域都表现出色。

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.

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 三体问题的研究中，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Topology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 waves (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Poincaré conjecture，随着时间的推移，这成为著名的悬而未决的数学问题之一，直到2002年至2003年被格里戈里·佩雷尔曼Grigori Perelman解决。

Life生平

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.

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，庞加莱出生在公爵城 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 莱翁·波因加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。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教育

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

拇指|右| 200px |南希市格兰德街117号房子亨利·彭加勒出生地的牌匾

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 · 庞加莱出生地的牌匾

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，受到他母亲欧热尼 · 劳诺伊斯(Eugénie Launois，1830-1897)的特别照料。

In 1862, Henri entered the Lycée in Nancy (now renamed the 模板:Ill 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.

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，亨利进入南希的莱西学院（现与亨利庞加莱大学一起为纪念他而改名，也在南希）。他在莱西大学呆了11年，在这段时间里，他证明了他所学的每一个科目都是尖子生之一。他擅长写作。他的数学老师称他为“数学怪兽”，他在法国所有莱卡顶尖学生的竞赛中获得一等奖。他最差的科目是音乐和体育，被描述为“充其量一般”。然而，视力差和心不在焉的倾向可能解释了这些困难。他于1871年毕业于莱西大学，获得文学和科学学士学位。

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.

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.

庞加莱于1873年以第一名的预选资格进入埃科尔综合理工学院École Polytechnique，并于1875年毕业。在那里，他作为查尔斯赫米特的学生学习数学，继续取得优异成绩，并于1874年发表了他的第一篇论文（“表面指示器性能的新示范”）。从1875年11月到1878年6月，他在埃科尔矿山学院学习，同时在采矿工程课程之外继续学习数学，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 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.

毕业于埃科尔矿业学院后，他加入了矿业兵团，担任法国东北部维苏尔地区的检查员。1879年8月，他在马格尼矿难现场，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正式调查。

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·赫米特的指导下，正在准备攻读数学科学博士学位。他的博士论文是关于微分方程的。它被命名为“关于部分差公式定义的函数属性”。庞加莱发明了一种研究这些方程性质的新方法。他不仅面临确定这类方程积分的问题，而且是第一个研究这些方程一般几何性质的人。他意识到，它们可以用来模拟太阳系内多个自由运动物体的行为。庞加莱于1879年毕业于巴黎大学。

The young Henri Poincaré

左|直立|拇指|年轻的亨利·庞加莱 The young Henri Poincaré

年轻的亨利 · 庞加莱

First scientific achievements最初科学成就

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.

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（1879年12月）担任数学初级讲师。同时，他发表了第一篇关于一类自守函数处理的重要文章。

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).

在卡昂，他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.

庞加莱立即跻身欧洲最伟大的数学家之列，吸引了许多著名数学家的注意。1881年，庞加莱被邀请到巴黎大学理学院担任教学职务；他接受了邀请。1883年至1897年间，他在埃科尔综合理工学院École Polytechnique教授 数学分析Mathematical analysis。

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职业生涯

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.

他从未完全放弃采矿业而投身于数学。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 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.

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, 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.)

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.)

1887年，他以解决有关多个轨道物体自由运动的三体问题，赢得了瑞典国王奥斯卡二世的数学竞赛。（参见下面的 三体问题Three-body problem部分。）

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 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.)

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，他加入了法国经度局French Bureau des Longitudes ,使他参与了世界各地时间的同步工作。1897年，庞加莱支持了一个不成功的建议，即循环尺度的十进制化，从而得到时间和经度。正是这篇文章促使他考虑建立国际时区的问题，以及相对运动的物体之间的时间同步问题。（参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 (SAF), the French astronomical society, from 1901 to 1903.

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

从1901年到1903年，庞加莱是法国天文学会（SAF）的主席。

Students学生

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

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 死亡

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.

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手术，随后于1912年7月17日在巴黎死于栓塞。时年58岁。他被安葬在巴黎蒙帕纳斯公墓的庞加莱家族墓穴中。

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.

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·阿雷格里（Claude Allègre）在2004年提议将庞加莱重新安葬在巴黎的潘太翁教堂（Panthéon），那里只为最高荣誉的法国公民保留。

Work工作

Summary综述

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.

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.

庞加莱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
• 代数拓扑
• the theory of analytic functions of several complex variables
• 多复变量|多复变量解析函数理论
• the theory of abelian functions
• [abelian变体|交换函数理论]]
• algebraic geometry
• 代数几何
• the Poincaré conjecture, proven in 2003 by Grigori Perelman.
• 庞加莱猜想，2003年由格里高里佩雷尔曼证明。
• Poincaré recurrence theorem
• 庞加莱递推定理
• hyperbolic geometry
• 双曲几何
• number theory
• 数论
• the three-body problem
• 三体问题
• the theory of diophantine equations
• 丢番图方程|丢番图方程理论
• electromagnetism
• 电磁学
• the special theory of relativity
• 狭义相对论
• the fundamental group
• 基本群
• 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 sphere and the Poincaré map.
• 在微分方程领域，庞加莱给出了许多对微分方程定性理论至关重要的结果，例如庞加莱球和庞加莱映射。
• Poincaré on "everybody's belief" in the Normal Law of Errors (see normal distribution for an account of that "law")
• 庞加莱关于“每个人的信仰”的q:Henri Poincaré‘正态误差定律’（参见正态分布中关于“法则”的解释）
• Published an influential paper providing a novel mathematical argument in support of quantum mechanics.
• 发表了一篇有影响力的论文，提供了一个新的数学论证来支持量子力学。

Three-body problem三体问题

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:

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此问题的方法的人。声明非常具体:

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.

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.

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，在假设没有两个质点相撞的情况下，找出每个质点的坐标在一个已知的时间函数的变量中的一个级数的表示，对该变量的所有值都是一致收敛的。

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 = 3 in 1912 and was generalised to the case of n > 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相对论部分的工作

Marie Curie and Poincaré talk at the 1911 Solvay Conference

[[图片：居里和庞加莱1911苏威.jpg|拇指|右|玛丽居里和1911年的庞加莱谈话索尔维会议]

Main articles: Lorentz ether theory and History of special relativity

{{main{洛伦兹以太理论{狭义相对论史}}

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]\displaystyle{ t^\prime = t-v x/c^2 \, }[/math]

在经度局建立国际时区的工作使庞加莱考虑如何使地球上静止的时钟（相对于绝对空间（或“以太Luminiferous aether”）以不同的速度移动）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”[math]\displaystyle{ t^\prime = t-v x/c^2 \, }[/math]

Local time本地时间

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 迈克尔逊·莫利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]\displaystyle{ t^\prime = t-v x/c^2 \, }[/math]

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.

“稍加反思就足以理解，所有这些肯定本身都没有意义。只有在约定成立的情况下，才能成立。”他还认为，科学家必须将光速的恒定性作为一个假设，以使物理理论具有最简单的形式。

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年对洛伦兹关于本地时间的“奇妙发明”进行了讨论，并指出，当移动的时钟通过交换假定在移动帧中以相同速度在两个方向上传播的光信号来同步时，就出现了这种情况。

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).

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, "

庞加莱一直是洛伦兹理论的解释者（有时是友好的批评家）。作为一个哲学家，庞加莱对“更深层的意义”很感兴趣。因此，他解释了洛伦兹的理论，并由此提出了许多与 狭义相对论相关的见解。在《时间的度量》(1898)中，庞加莱说

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.

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.

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相对论原理与洛伦兹变换

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Principle of relativity，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]\displaystyle{ 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).

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Time dilation factor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所知的相对论速度加法定律Relativistic velocity-addition law。

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.

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

[math]\displaystyle{ 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] 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 >

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. 并于1904年将其命名为相对论，根据这一原理，任何物理实验都无法区分均匀运动状态和静止状态。

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.

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确实是正确的，毕竟要使洛伦兹变换形成一个群，他还给出了现在所知的相对论速度加法定律。

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:

庞加莱后来在1905年6月5日巴黎科学院会议上发表了一篇论文，其中讨论了这些问题。在出版的版本中，他这样写道：

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。能量携带质量和用有争议的乙太解决方案来弥补上述问题的可能性the possibility that energy carries mass and criticized the ether solution to compensate the above-mentioned problems

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]\displaystyle{ 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]

洛伦兹建立的基本点是，电磁场的方程不会因某种形式的变换（我称之为洛伦兹）而改变：

[math]\displaystyle{ 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]

He also discussed two other unexplained effects: (1) non-conservation of mass implied by Lorentz's variable mass [math]\displaystyle{ \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[math]\displaystyle{ \gamma m }[/math]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]\displaystyle{ \ell\left(\varepsilon\right) }[/math] must be unity for all [math]\displaystyle{ \varepsilon }[/math] (Lorentz had set [math]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ 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.

庞加莱在1907年表示对他的新力学的四维重新表述缺乏兴趣，因为在他看来，将物理学翻译成四维几何的语言需要付出太多的努力才能获得有限的利润。所以1907年由Hermann Minkowski提出了这个概念的结果。

Mass–energy relation质量-能量关系

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，亨利·庞加莱首次提出引力波（ondes gravifiques），它从物体发出并以光速传播。在公开场合，爱因斯坦在1921年发表的一篇演讲中承认了庞加莱的存在，他在演讲中称之为几何与非欧几里德几何有关，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命名的变换对于分析麦克斯韦方程组是必不可少的，而庞加莱进一步深化了这一见解……”

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. 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/c². 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.

像以前的其他一样，庞加莱（1900）发现了质量和电磁能之间的关系。在研究作用/反应原理和洛伦兹以太理论之间的冲突时，他试图确定当包含电磁场时，重心是否仍以匀速运动。他注意到作用/反作用原理不仅适用于物质，而且电磁场有其自身的动量。庞加莱得出结论，电磁波的电磁场能量表现为一个虚拟的流体（“流体虚拟”），质量密度为E/c²。如果质心框架由物质的质量和虚拟流体的质量共同定义，并且如果虚拟流体是不可摧毁的，它既不会被创造也不会被摧毁，那么质量中心框架的运动保持一致。但是电磁能可以转化成其他形式的能量。因此，庞加莱假设在空间的每一点都存在一个非电能流体，它可以将电磁能转化为它，它也携带着与能量成比例的质量。这样，质心的运动保持一致。庞加莱说，人们不应该对这些假设感到太惊讶，因为它们只是数学上的虚构。

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.

然而，庞加莱的解决方案导致了一个悖论：如果 赫兹振子朝某个方向辐射，它将受到虚拟流体惯性的反冲。庞加莱对移动源的帧执行了洛伦兹升压Lorentz boost（顺序为“v”/“c”）。他指出，能量守恒在这两个框架中都成立，但动量守恒定律被违反了。这就允许了永动机，一个他深恶痛绝的概念。自然法则必须在参照系中有所不同，相对论原理就不成立了。因此，他认为，在这种情况下， 乙太中必须有另一种补偿机制。

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 本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 本地时间。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，发展了一个与数学等价的运动学。

Poincaré himself came back to this topic in his St. Louis lecture (1904). This time (and later also in 1908) he rejected the possibility that energy carries mass and criticized the ether solution to compensate the above-mentioned problems:

庞加莱本人在圣路易斯讲座（1904）中又回到了这个话题上。这次（后来也是在1908年），他拒绝了米勒1981年出版的《相对论的第二资源：能量携带质量的可能性》，并批评了以太方案来补偿上述问题：

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，庞加莱和洛伦兹是真正的相对论发现者。

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{{引用}这个装置会后坐，就像它是一门大炮，投射的能量是一个球，这与牛顿原理相矛盾，因为我们现在的弹丸没有质量；它不是物质，而是能量。[…]我们是否可以说，将振荡器与接收器分开的空间，以及扰动在从一个到另一个的过程中必须穿过的空间并不是空的，而是不仅充满了乙太，而且充满了空气，甚至在行星间的空间中，充满了一些微妙而可测量的流体；这个物质受到了冲击，就像接收器，在能量到达它的那一刻，当干扰离开它时，它会反冲？这可以拯救牛顿原理，但事实并非如此。如果能量在它的传播过程中始终依附于某些物质的底层，这种物质就会携带着光，而菲佐已经证明，至少对空气来说，没有这种物质。迈克尔逊和莫利后来证实了这一点。我们也可以假设物质本身的运动完全被以太的运动所补偿，但这会使我们产生与刚才一样的考虑。这个原理，如果这样解释的话，可以解释任何东西，因为不管是什么可见的运动，我们都可以想象出假想的运动来补偿它们。但如果它能解释任何事情，它将使我们什么也不能预言；它将不允许我们在各种可能的假设之间作出选择，因为它预先解释了一切。因此，它变得毫无用处。}}

He also discussed two other unexplained effects: (1) non-conservation of mass implied by Lorentz's variable mass [math]\displaystyle{ \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影响：（1）洛伦兹的变质量[math]\displaystyle{ \gamma m }[/math]所暗示的质量非守恒性，亚伯拉罕的变质量理论和 考夫曼关于快速移动电子质量的实验和（2）居里夫人镭实验中的能量非守恒。

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。

Topological transformation of the torus into a mug 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 = E/c² 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.

正是阿尔伯特·爱因斯坦的质能守恒（1905年）的概念，一个以辐射或热的形式损失能量的物体，其质量损失量为m = E/c² 解决了庞加莱悖论，没有使用以太内部的任何补偿机制。 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.

Gravitational waves引力波

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 费利克斯·克莱因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 恩里科·贝蒂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. "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."

1905年，亨利·庞加莱首次提出了由物体发出并以光速传播的引力波（“ondes graviques”）。“重要的一点是，检查者必须对重力的作用进行修正。“这是一个假设万有引力传播的管道，它是地球引力传播的一个假设，它是地球引力的一个重要组成部分。”

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Homotopy and Homology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 贝蒂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庞加莱和爱因斯坦

Einstein's first paper on relativity was published three months after Poincaré's short paper, but before Poincaré's longer version. 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. 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 ...."

爱因斯坦关于相对论的第一篇论文发表在庞加莱的短篇论文《1905年论文》发表三个月之后，但在庞加莱的长篇论文发表之前。爱因斯坦依靠相对论原理推导出洛伦兹变换，并使用了类似的时钟同步程序（爱因斯坦同步）庞加莱（1900年）曾描述过，但爱因斯坦的论文很了不起，因为它根本没有参考文献。庞加莱从未承认爱因斯坦在狭义相对论上的工作。然而，爱因斯坦在1919年5月3日写给汉斯-瓦因格的信中对庞加莱的观点表示了认同的倾向，当时爱因斯坦认为瓦辛格的总体观点接近于他自己，而庞加莱则接近瓦辛格。 在公开场合，爱因斯坦在1921年的一次演讲中追认了庞加莱，他的演讲名为“几何与Erfahrung”，与非欧几里德几何有关，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命名的变换对于分析麦克斯韦方程组是必不可少的，而庞加莱进一步深化了这一见解……”

Chaotic motion in three-body problem (computer simulation). 三体问题的混乱运动(计算机模拟)

Assessments on Poincaré and relativity对庞加莱和相对论的评价

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 布鲁斯Bruns (1887)的理论进行概括，庞加莱 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。

模板:Further {{进一步{狭义相对论史{相对论优先权争议}}

Poincaré's work in the development of special relativity is well recognised, though most historians stress that despite many similarities with Einstein's work, the two had very different research agendas and interpretations of the work. 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.

庞加莱在狭义相对论的发展中的工作是公认的，大多数历史学家强调，尽管与爱因斯坦的工作有许多相似之处，但两人的研究议程和对这项工作的解释截然不同。 。与爱因斯坦相反，他在论文中继续使用以太的概念，认为以太静止时显示“真实”时间，而移动的时钟显示本地时间。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，发展了一种与数学上等价的运动学。

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.

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

Algebra and number theory代数与数论

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

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) Singular points (saddle, focus, center, node)的分类，引入了 极限环和环指数Limit cycle and Loop index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Integral invariants和 变分方程Variational equations解的一般理论。对于有限差分方程 Finite-difference equations，他创造了一个新的方向——解的 渐近分析Asymptotic analysis。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. He also made major contributions to the theory of discrete groups and their representations. 庞加莱把群论引入物理学，是第一个研究[[洛伦兹变换]群的人。 他还对离散群理论及其表示法做出了重大贡献。

Topological transformation of the torus into a mug

右| frame | 50px |

圆环到杯子的拓扑变换

• The singular points of the integral curves
• 

  Saddle

• 

  Saddle

• ===Topology拓扑学===
• 

  Focus

• 

  Focus

• 

  Center

• Phase portrait Center.svg

  Center

• 

  Node

• 

  Node

</gallery >

Astronomy and celestial mechanics天文学与天体力学

Chaotic motion in three-body problem (computer simulation).

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.

庞加莱出版了两本经典专著《天体力学的新方法》（1892-1899）和《天体力学讲座》（1905-1910）。其中，他成功地将他们的研究成果应用于三体的运动问题，并详细研究了解的行为（频率、稳定性、渐近性等）。介绍了小参数法、不动点、积分不变量、变分方程、渐近展开的收敛性。推广Bruns（1887）的一个理论，庞加莱证明了 三体问题是不可积的。换言之， 三体问题的一般解不能通过物体的明确坐标和速度用代数函数和超越函数来表示。他在这方面的工作是自艾萨克牛顿以来在天体力学方面的第一个重大成就。

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 systems.

这些专著包括了庞加莱的思想，这后来成为数学“混沌理论”的基础（特别参见庞加莱递推定理）和动力系统s的一般理论。

The mathematician Darboux claimed he was un intuitif (an intuitive), arguing that this is demonstrated by the fact that he worked so often by visual representation. He did not care about being rigorous and disliked logic. (Despite this opinion, Jacques Hadamard wrote that Poincaré's research demonstrated marvelous clarity and Poincaré himself wrote that he believed that logic was not a way to invent but a way to structure ideas and that logic limits ideas.)

数学家 Darboux 声称他是反直觉的(一种直觉) ，认为这由他工作如此频繁的视觉表示的事实可以证明。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地 -- 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).

庞加莱为引力旋转流体的平衡图写了重要的天文学著作。他引入了 分支点的重要概念，证明了非椭球体（包括环形和梨形）等平衡图形的存在性及其稳定性。这项天文发现奖（1900年）被英国皇家天文学会授予。

Differential equations and mathematical physics微分方程与数学物理

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). 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.

在为他关于微分方程组奇点研究的博士论文辩护后，庞加莱以“微分方程定义的曲线”（1881-1882）为题写了一系列回忆录。在这些文章中，他建立了一个新的数学分支，叫做“微分方程定性理论”。Poincaré表明，即使微分方程不能用已知函数来求解，但是从方程的形式来看，可以找到关于解的性质和行为的丰富信息。特别地，Poincaré研究了积分曲线在平面上的轨迹性质，给出了奇异点（鞍点、焦点、中心、节点）的分类，引入了极限环和环指数的概念，证明了除某些特殊情况外，极限环的个数始终是有限的。庞加莱还发展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，他开创了一个新的方向——解的渐近分析。他将这些成果应用于研究数学物理和天体力学的实际问题，所采用的方法是其拓扑学工作的基础。

• The singular points of the integral curves
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  Saddle

• 

  Focus

• 

  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:
• 

  Node

Character

文件:Henri Poincaré by H Manuel.jpg

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. (Despite this opinion, Jacques Hadamard wrote that Poincaré's research demonstrated marvelous clarity and Poincaré himself wrote that he 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). 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 a.m. and noon then again from 5 p.m. to 7 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:

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Named after him

以他的名字命名

Attitude towards transfinite numbers

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.

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

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

< 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)

• 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 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. 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. 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).

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. 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.

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 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。

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On celestial mechanics:

关于天体力学:

Poincaré believed that arithmetic is 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 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 impredicative definitions.

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 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". 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.

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.

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

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.

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

Bibliography

Other:

其他:

Poincaré's writings in English translation

Popular writings on the philosophy of science:

• Poincaré, Henri

Exhaustive bibliography of English translations:

详尽的英语翻译书目: (1902–1908), The Foundations of Science, New York: Science Press {{citation}}: line feed character in |author= at position 16 (help)CS1 maint: extra punctuation (link); 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. Last Essays., New York: Dover reprint, 1963

• 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. Analysis Situs (PDF). The first systematic study of topology.

On celestial mechanics:

• 1892–99. New Methods of Celestial Mechanics, 3 vols. English trans., 1967.
  .
  • 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:

  • 1892–2017. Henri Poincaré Papershttps://en.wikipedia.org/wiki/Defekte_Weblinks?dwl={{{url}}} Seite nicht mehr abrufbar], Suche in Webarchiven: Kategorie:Wikipedia:Weblink offline (andere Namensräume)[http://timetravel.mementoweb.org/list/2010/Kategorie:Wikipedia:Vorlagenfehler/Vorlage:Toter Link/URL_fehlt .

  
  

  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

• Poincaré homology sphere

• Poincaré inequality

• Poincaré map

• Poincaré residue

• Poincaré series (modular form)

• Poincaré space

• Poincaré metric

• Poincaré plot

• Poincaré series

• 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

• 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

References

Footnotes

Sources

• Bell, Eric Temple, 1986. Men of Mathematics (reissue edition). Touchstone Books.
  .
  • Belliver, André, 1956. Henri Poincaré ou la vocation souveraine. Paris: Gallimard.
  • Bernstein, Peter L, 1996. "Against the Gods: A Remarkable Story of Risk". (p. 199–200). John Wiley & Sons.
  • Boyer, B. Carl, 1968. A History of Mathematics: Henri Poincaré, John Wiley & Sons.
  • Grattan-Guinness, Ivor, 2000. The Search for Mathematical Roots 1870–1940. Princeton Uni. Press.
  • Dauben, Joseph (2004) [1993], "Georg Cantor and the Battle for Transfinite Set Theory" (PDF), Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, CA), pp. 1–22, archived from the original (PDF) on 13 July 2010. Internet version published in Journal of the ACMS 2004.
  • Folina, Janet, 1992. Poincaré and the Philosophy of Mathematics. Macmillan, New York.
  • Gray, Jeremy, 1986. Linear differential equations and group theory from Riemann to Poincaré, Birkhauser
    • Gray, Jeremy, 2013. Henri Poincaré: A scientific biography. Princeton University Press
      • Jean Mawhin (October 2005), "Henri Poincaré. A Life in the Service of Science" (PDF), Notices of the AMS, 52 (9): 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.
      • Peterson, Ivars, 1995. Newton's Clock: Chaos in the Solar System (reissue edition). W H Freeman & Co.
        .
        • 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.
        • Stillwell, John (2010). Mathematics and Its History (3rd, illustrated ed.). Springer Science & Business Media. ISBN 978-1-4419-6052-8. https://books.google.com/books?id=V7mxZqjs5yUC. 
        • 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.
        • 模板:PlanetMath attribution

        
        

        Further reading

        Secondary sources to work on relativity

        • Cuvaj, Camillo (1969), "Henri Poincaré's Mathematical Contributions to Relativity and the Poincaré Stresses", American Journal of Physics, 36 (12): 1102–1113, Bibcode:1968AmJPh..36.1102C, doi:10.1119/1.1974373
        • Darrigol, O. (1995), "Henri Poincaré's criticism of Fin De Siècle electrodynamics", Studies in History and Philosophy of Science, 26 (1): 1–44, Bibcode:1995SHPMP..26....1D, doi:10.1016/1355-2198(95)00003-C
        • Darrigol, O. (2000), Electrodynamics from Ampére to Einstein, Oxford: Clarendon Press, ISBN 978-0-19-850594-5
        • Darrigol, O. (2004), "The Mystery of the Einstein–Poincaré Connection", Isis, 95 (4): 614–626, doi:10.1086/430652, PMID 16011297, S2CID 26997100
        • Darrigol, O. (2005), "The Genesis of the theory of relativity" (PDF), Séminaire Poincaré, 1: 1–22, Bibcode:2006eins.book....1D, doi:10.1007/3-7643-7436-5_1, ISBN 978-3-7643-7435-8
        • Galison, P. (2003), Einstein's Clocks, Poincaré's Maps: Empires of Time, New York: W.W. Norton, ISBN 978-0-393-32604-8
        • Giannetto, E. (1998), "The Rise of Special Relativity: Henri Poincaré's Works Before Einstein", Atti del XVIII Congresso di Storia della Fisica e dell'astronomia: 171–207
        • Giedymin, J. (1982), Science and Convention: Essays on Henri Poincaré's Philosophy of Science and the Conventionalist Tradition, Oxford: Pergamon Press, ISBN 978-0-08-025790-7
        • Goldberg, S. (1967), "Henri Poincaré and Einstein's Theory of Relativity", American Journal of Physics, 35 (10): 934–944, Bibcode:1967AmJPh..35..934G, doi:10.1119/1.1973643
        • Goldberg, S. (1970), "Poincaré's silence and Einstein's relativity", British Journal for the History of Science, 5: 73–84, doi:10.1017/S0007087400010633
        • Holton, G. (1988) [1973], "Poincaré and Relativity", Thematic Origins of Scientific Thought: Kepler to Einstein, Harvard University Press, ISBN 978-0-674-87747-4
        • Katzir, S. (2005), "Poincaré's Relativistic Physics: Its Origins and Nature", Phys. Perspect., 7 (3): 268–292, Bibcode:2005PhP.....7..268K, doi:10.1007/s00016-004-0234-y, S2CID 14751280
        • Keswani, G.H., Kilmister, C.W. (1983), "Intimations of Relativity: Relativity Before Einstein", Br. J. Philos. Sci., 34 (4): 343–354, doi:10.1093/bjps/34.4.343, archived from the original on 26 March 2009{{citation}}: CS1 maint: multiple names: authors list (link)
        • Keswani, G.H. (1965), "Origin and Concept of Relativity, Part I", Br. J. Philos. Sci., 15 (60): 286–306, doi:10.1093/bjps/XV.60.286
        • Keswani, G.H. (1965), "Origin and Concept of Relativity, Part II", Br. J. Philos. Sci., 16 (61): 19–32, doi:10.1093/bjps/XVI.61.19
        • Keswani, G.H. (1966), "Origin and Concept of Relativity, Part III", Br. J. Philos. Sci., 16 (64): 273–294, doi:10.1093/bjps/XVI.64.273
        • Kragh, H. (1999), Quantum Generations: A History of Physics in the Twentieth Century, Princeton University Press, ISBN 978-0-691-09552-3
        • Langevin, P. (1913), "L'œuvre d'Henri Poincaré: le physicien", Revue de Métaphysique et de Morale, 21: 703
        • Macrossan, M. N. (1986), "A Note on Relativity Before Einstein", Br. J. Philos. Sci., 37 (2): 232–234, CiteSeerX 10.1.1.679.5898, doi:10.1093/bjps/37.2.232, archived from the original on 29 October 2013, retrieved 27 March 2007
        • Miller, A.I. (1973), "A study of Henri Poincaré's "Sur la Dynamique de l'Electron", Arch. Hist. Exact Sci., 10 (3–5): 207–328, doi:10.1007/BF00412332, S2CID 189790975
        • Miller, A.I. (1981), Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911), Reading: Addison–Wesley, ISBN 978-0-201-04679-3
        • Miller, A.I. (1996), "Why did Poincaré not formulate special relativity in 1905?", in Jean-Louis Greffe; Gerhard Heinzmann; Kuno Lorenz (eds.), Henri Poincaré : science et philosophie, Berlin, pp. 69–100
        • Schwartz, H. M. (1971), "Poincaré's Rendiconti Paper on Relativity. Part I", American Journal of Physics, 39 (7): 1287–1294, Bibcode:1971AmJPh..39.1287S, doi:10.1119/1.1976641
        • Schwartz, H. M. (1972), "Poincaré's Rendiconti Paper on Relativity. Part II", American Journal of Physics, 40 (6): 862–872, Bibcode:1972AmJPh..40..862S, doi:10.1119/1.1986684
        • Schwartz, H. M. (1972), "Poincaré's Rendiconti Paper on Relativity. Part III", American Journal of Physics, 40 (9): 1282–1287, Bibcode:1972AmJPh..40.1282S, doi:10.1119/1.1986815
        • Scribner, C. (1964), "Henri Poincaré and the principle of relativity", American Journal of Physics, 32 (9): 672–678, Bibcode:1964AmJPh..32..672S, doi:10.1119/1.1970936
        • Walter, S. (2005), "Henri Poincaré and the theory of relativity", in Renn, J. (ed.), Albert Einstein, Chief Engineer of the Universe: 100 Authors for Einstein, Berlin: Wiley-VCH, pp. 162–165
        • Walter, S. (2007), "Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905–1910", in Renn, J. (ed.), The Genesis of General Relativity, vol. 3, Berlin: Springer, pp. 193–252
        • Whittaker, E.T. (1953), "The Relativity Theory of Poincaré and Lorentz", A History of the Theories of Aether and Electricity: The Modern Theories 1900–1926, London: Nelson
        • Zahar, E. (2001), Poincaré's Philosophy: From Conventionalism to Phenomenology, Chicago: Open Court Pub Co, 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

        • Leveugle, J. (2004), La Relativité et Einstein, Planck, Hilbert—Histoire véridique de la Théorie de la Relativitén, Pars: L'Harmattan

        Category:20th-century French philosophers

        范畴: 20世纪法国哲学家

        • Logunov, A.A. (2004), Henri Poincaré and relativity theory, 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

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        • Internet Encyclopedia of Philosophy: "Henri Poincaré"—by Mauro Murzi.

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        • Internet Encyclopedia of Philosophy: "Poincaré’s Philosophy of Mathematics"—by Janet Folina.

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        类别: 法国物理学家

        • 亨利·庞加莱 Jules Henri Poincaré at the Mathematics Genealogy Project

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        • Henri Poincaré on Information Philosopher

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        类别: 数学分析师

        • 模板:MacTutor Biography

        Category:Members of the Académie Française

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        • 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

        • Henri Poincaré Papers University of Nantes (in French).

        Category:Mines ParisTech alumni

        类别: Mines ParisTech alumni

        • Bruce Medal page

        Category:Officers of the French Academy of Sciences

        类别: 法国科学院官员

        • Collins, Graham P., "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, "Discussion of the Poincaré conjecture," 2 November 2006, hosted by Melvynn Bragg.

        Category:Philosophers of science

        范畴: 科学哲学家

        • Poincare Contemplates Copernicus at MathPages

        Category:Recipients of the Bruce Medal

        类别: 布鲁斯奖章获得者

        • High Anxieties – The Mathematics of Chaos (2008) BBC documentary directed by 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

        Category:Thermodynamicists

        类别: 热力学家

        Category:Fluid dynamicists

        类别: 流体动力学家

        Category:Topologists

        类别: 拓扑学家

        Category:University of Paris faculty

        类别: 巴黎大学教员

        Category:French male writers

        类别: 法国男性作家

        Category:Deaths from embolism

        类别: 死于栓塞

        Category:Dynamical systems theorists

        范畴: 动力系统理论家

        
        

        This page was moved from wikipedia:en:Henri Poincaré. Its edit history can be viewed at 庞加莱/edithistory