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{{#seo:
|keywords=亨利·庞加莱 Henri Poincaré，复杂性理论，复杂性经济学，自主经济学
|description=庞加莱猜想，三体问题，拓扑学，狭义相对论
}}
== 基本信息 ==
[[File:Henri Poincare.jpg|230px|left]]

{| class="wikitable" width="50%"
|-
! 类别 !! 信息
|-
| 姓名 || 亨利·庞加莱 Jules Henri Poincaré
|-
| 出生日期 || 1854年4月29日
|-
| 出生地 || Nancy,Lorraine,France 南希，洛林，法国
|-
|居住地 || France 法国
|-
| 所在机构 || 法国矿业团，卡昂大学，巴黎大学，法国经度理局
|-
| 主要贡献 || 庞加莱猜想，三体问题，拓扑学，狭义相对论，庞加莱－霍普夫定理，庞加莱对偶性，庞加莱–伯克霍夫–威特定理，庞加莱不等式，希尔伯特–庞加莱级数，庞加莱度量，旋转数，提出术语贝蒂数，分岔理论，混沌理论，布劳威尔不动点定理，球体领域，庞加莱－本迪克松定理，庞加莱－林德斯泰特方法，庞加莱复现定理，庞加莱圆盘模型
｜—
|}

 

== 个人介绍 ==
 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。

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。

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。

物理和数学方面的庞加莱小组就是以他的名字命名的。

在20世纪早期，他制定了 庞加莱猜想Poincaré conjecture，随着时间的推移，这成为著名的数学难题之一，直到2002年至2003年被格里戈里·佩雷尔曼Grigori Perelman解决。

== 教育经历 ==
* 1862年-1871年，莱西学院，文学和科学学士
* 1873年-1875年，埃科尔综合理工学院
* 1875年-1878年，埃科尔矿山学院采矿工程/数学
* 1879年，巴黎大学，数学科学博士

== 任职经历 ==

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。

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。

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。

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。

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。

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。

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。）

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。）

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。

从1901年到1903年，庞加莱是法国天文学会（SAF）的主席。

== 学术背景 ==
=== 研究方向 ===

===Summary综述===
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。

他还是数学和物理的科普工作者，并为普通大众写了几本书。
他提出的具体主题包括:

*[[algebraic topology]]
*[[代数拓扑]]
*[[several complex variables|the theory of analytic functions of several complex variables]]
*多复变量|多复变量解析函数理论
*[[abelian variety|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]]
*[[diophantine equation|the theory of diophantine equations]]
*丢番图方程|丢番图方程理论
*[[electromagnetism]]
*[[电磁学]]
*[[Special relativity|the special theory of relativity]]
*狭义相对论
*the [[fundamental group]]
*[[基本群]]
*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[[庞加莱同调球|庞加莱球]]和[[庞加莱映射]]
*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）
*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]]。<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三体问题===

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:

<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。</blockquote >

如果这个问题无法解决，那么对经典力学的任何其他重要贡献都将被认为能够获奖。虽然庞加莱没有解决最初的问题，但最终还是把奖颁给了他。其中一位评委，著名的卡尔·魏尔斯特拉斯说：“这项工作确实不能被视为提供了所提出问题的完整解决方案，但它的出版将开创天体力学史上的一个新纪元。”详细内容见格林的一篇文章。最终印刷的版本包含了许多导致混沌理论的重要思想。最初所述的问题最终由Karl F.Sundman在1912年解决了n = 3的情况，并在1990年代将其推广到王秋东的n > 3体的案例中。

===Work on relativity 相对论部分的工作===

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>t^\prime = t-v x/c^2 \,</math>

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)。

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“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。“

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。1881年，庞加莱用 双曲面模型Hyperboloid model描述了 双曲几何学Hyperbolic geometry，提出了洛伦兹区间<math>x^2+y^2-z^2=-1</math>上不变的变换，使其在数学上等价于2+1维的 洛伦兹变换。此外，庞加莱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 贝尔特拉米-克莱因Beltrami–Klein模型都可以与相对论速度空间（见 陀螺矢量空间）相关。

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====

他在1900年的两篇论文中讨论了“相对运动原理”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。

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。

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。

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: 洛伦兹建立的基本观点是，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。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。<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>

1905年，庞加莱写信给洛伦兹，谈到洛伦兹1904年的论文，这篇论文被庞加莱称为“最重要的论文”。在这封信中，他指出了洛伦兹在将其变换应用于麦克斯韦方程组（电荷占据空间）时犯下的一个错误，并对洛伦兹给出的时间膨胀因子提出了质疑。<ref name="univ-nantes">

并证明了任意函数<math>\ell\left(\varepsilon\right)</math>对于所有<math>\varepsilon</math>必须是统一的（Lorentz通过一个不同的参数设置<math>\ell = 1</math>），以使变换形成一个组。在1906年发表的论文的放大版中，庞加莱指出组合<math>x^2+ y^2+ z^2- c^2t^2</math>是不变的。他通过引入<math>ct\sqrt{-1}</math>作为第四个虚坐标，指出Lorentz变换仅仅是四维空间中绕原点的旋转，他使用了四个向量的早期形式。庞加莱在1907年表示对他的新力学的四维重新表述不感兴趣，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益处。1907年，由赫尔曼·明科夫斯基（Hermann Minkowski）得出了这个概念的后果。

{{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>

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。<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>

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。<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>

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

::<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>

<blockquote>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>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>

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>\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。并证明了任意函数<math>\ell\left（\varepsilon\right）</math>对于所有<math>\varepsilon</math>必须是统一的（Lorentz通过一个不同的参数设置<math>\ell=1</math>），以使变换形成一个组。在1906年发表的论文的放大版中，庞加莱指出组合<math>x^2+y^2+z^2-c^2t^2</math>是[[不变量（数学）|不变量]]。他指出，通过引入<math>ct\sqrt{-1}</math>作为第四个虚坐标，Lorentz变换仅仅是四维空间中绕原点的旋转，他使用了[[四向量]]s的早期形式。<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

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Hertzian oscillato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。

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。<ref>Walter (2007), Secondary sources on relativity</ref>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====

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……”

像以前的[[质量-能量等效性#电磁静止质量|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[[牛顿运动定律|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。<ref name=action/>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''/''c''2。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。

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。

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。

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。

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。

====Gravitational waves引力波====

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。

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”）。<ref name=“1905 paper”/>“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。”

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====

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。<ref name=long/>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。<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}}另见本函及其评注{{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> 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……”<ref>Darrigol 2004, Secondary sources on relativity</ref>

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

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。

{{Further|History of special relativity|Relativity priority dispute}}
{{进一步{狭义相对论史{相对论优先权争议}}

庞加莱在狭义相对论的发展中的工作是公认的，<ref name=darrigol />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。<ref>Galison 2003 and Kragh 1999, Secondary sources on relativity</ref> 。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。

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。

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。

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

===Algebra and number theory代数与数论===

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。

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。<ref>Poincaré, Selected works in three volumes. page = 682{{full citation needed|date=September 2019}}</ref> 他还对离散群理论及其表示法做出了重大贡献。

===Topology拓扑学===

[[费利克斯·克莱因]]在他的“Erlangen程序”（1872）中明确地定义了这个主题：任意连续变换的几何不变量，一种几何学。正如[[约翰本尼迪克特名单]]所建议的，引入了术语“拓扑”，而不是以前使用的“分析位置”。一些重要的概念是由[[恩里科·贝蒂]]和[[伯恩哈德·黎曼]]介绍的。但这一科学的基础，对于任何维度的空间，都是由庞卡莱创造的。他关于这个主题的第一篇文章发表在1894年。

他对几何学的研究导致了[[同伦]]和[[同伦（数学）|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，如贝蒂Betti数和[[基本群]]。Poincaré证明了n维多面体的边数、顶点数和面数的公式（Euler-Poincaré定理），给出了维数直观概念的第一个精确表达式。<ref>{{citation|last=Aleksandrov|first=Pavel S. |authorlink=Pavel Alexandrov|title= Poincaré and topology| pages = 27–81}}{{full citation needed|date=September 2019}}</ref>

===Astronomy and celestial mechanics天文学与天体力学===

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

Photographic portrait of H. Poincaré by Henri Manuel

庞加莱肖像摄影: Henri Manuel

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。<ref>J. Stillwell, Mathematics and its history, [https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA254 page 254]</ref>

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。

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]]的一般理论

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。<ref>A. Kozenko, The theory of planetary figures, pages = 25–26{{full citation needed|date=September 2019}}</ref>

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

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。<ref>French: "Mémoire sur les courbes définies par une équation différentielle"</ref>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。<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>

==Character特征==

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)。

庞加莱的工作习惯被比作蜜蜂从一朵花飞到另一朵花。庞加莱对他的思维方式很感兴趣；他研究了自己的习惯，并于1908年在巴黎普通心理学研究所发表了一篇关于他的观察结果的演讲。他把自己的思维方式与他如何取得几个发现联系在一起。

他的思维方式可以很好地概括为: {{quote|text=“他习惯于忽略细节，只看山顶，以惊人的速度从一个瞬间到另一个瞬间，发现的事实围绕着他们的中心聚集在一起，并自动地被归入他的记忆中。”|sign=Belliver (1956)}}

数学家达布克斯声称他是“不直觉的”（直觉的），认为这一点可以从他经常通过视觉表现来工作的事实中得到证明。他不在乎严谨，也不喜欢逻辑。<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> （尽管如此，[[Jacques Hadamard]]写道，庞加莱的研究显示出惊人的清晰性。<ref>J. Hadamard. L'oeuvre de H. Poincaré. Acta Mathematica, 38 (1921), p. 208</ref>庞加莱自己也写道他<！--TODO:加上庞加莱对严谨的看法，参见http://www.forgottenbooks.org/readbook/American_Journal_of_Mathematics_1890_v12_1000084889#233 —每次我能做到的时候，我都是绝对严谨的-->相信逻辑不是一种发明的方式，而是一种构建思想的方式，逻辑限制了思想。）

庞加莱的心理组织不仅令庞加莱本人感兴趣，而且巴黎高等研究院心理学实验室的心理学家埃杜阿尔德·图卢兹也感兴趣。图卢兹写了一本书，名叫《亨利·庞加莱》（1910年）<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>在其中，他讨论了庞加莱的常规日程安排：

他每天在同一时间短时间内工作。他每天进行四个小时的数学研究，从上午10点到中午，然后再从下午5点到晚上7点。。晚上晚些时候他会在杂志上读文章。

他通常的工作习惯是在脑子里彻底解决一个问题，然后把完成的问题写在纸上。

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。”

他两手灵巧，近视。

当他去听课时，他对所听到的东西进行视觉化的能力被证明是特别有用的，因为他的视力很差，他不能正确地看到讲课者在黑板上写的东西。

这些能力在某种程度上被他的缺点所抵消：

*他身体笨拙，艺术上拙劣。

*他总是匆匆忙忙的，不喜欢回去修改或更正。

*他从来没有在一个问题上花很长时间，因为他相信潜意识会在他有意识地处理另一个问题时继续工作。

===Attitude towards transfinite numbers对超限数的态度===

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。庞加莱说：“没有真正的无限；坎托利亚人忘记了这一点，这就是他们陷入矛盾的原因。”<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

==科普著作==

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。

== 荣誉成就 ==
* 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。

*奥斯卡二世，瑞典数学竞赛之王（1887）
*[[美国哲学学会]]1899年
*伦敦（1900年）的[[皇家天文学会金牌]]
*1905 [博利奖]
* [ [马图基奖章] ] 1905
*[[法国科学院]]1906年
*〔〔阿克米-弗兰〕〕1909
＊[〔布鲁斯奖章〕〕（1911）

以他命名的

*[[亨利庞加莱学院]]（数学与理论物理中心）
*月球上的陨石坑 [[庞加莱（陨石坑)|庞加莱]]
*小行星

==Philosophy哲学==

关于代数拓扑:

庞加莱的哲学观点与[[伯特兰·罗素]]和[[哥特罗布·弗雷格]]相反，他们认为数学是[[逻辑]]的一个分支。庞加莱强烈反对，声称[[直觉（知识）|直觉]]是数学的生命。庞加莱在他的书“[[科学与假设]]”中提出了一个有趣的观点：{{quote | text=对于一个肤浅的观察者来说，科学的真理是不容置疑的；科学的逻辑是绝对正确的，如果科学家有时是错误的，这仅仅是因为他们错误地理解了它的规则}}

关于天体力学:

庞加莱认为[[算术]]是[[分析/合成区别|合成]]。他认为[[皮亚诺的公理]]不能用归纳法原理进行非循环证明（Murzi，1998），因此得出结论认为算术是“综合的而非分析的”。庞加莱接着说，数学不能从逻辑中推导出来，因为它不是分析性的。他的观点与[[Immanuel Kant]]的观点相似（Kolak，2001，Folina 1992）。他强烈反对Cantorian[[set theory]]，反对其使用[[不精确性|非指示性]]定义{{Citation needed|date=March 2018}}。

然而，庞加莱并没有在哲学和数学的所有分支中分享康德的观点。例如，在几何学中，庞加莱认为[[非欧几里德几何|非欧几里德空间]]的结构可以通过分析得到。庞加莱认为传统在物理学中起着重要的作用。他的观点（以及后来一些更极端的版本）被称为“[[传统主义]]”。<ref>Yemima Ben-Menahem, ''Conventionalism: From Poincare to Quine'', Cambridge University Press, 2006, p. 39.</ref>庞加莱认为[[牛顿第一定律]]不是经验性的，而是力学的传统框架假设（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>他还认为物理空间的几何学是传统的。他考虑了物理场的几何结构或温度梯度可以改变的例子，要么将一个空间描述为由刚性标尺测量的非欧几里德空间，要么描述为标尺通过可变热分布而膨胀或收缩的欧几里德空间。然而，庞加莱认为我们太习惯了[[欧几里德几何]]，我们宁愿改变物理定律来保存欧几里德几何，而不是转向非欧几里德物理几何。<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>

===Free will自由意志===

庞加莱在巴黎心理学学会之前的著名演讲（出版为“[[科学与假设]]”、[[科学的价值]]”和“科学与方法”）被[[Jacques Hadamard]]引用为创意和发明由两个心理阶段组成的思想来源，首先是可能的解决方案的随机组合一个问题，然后是一个批判性的评估<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]].
尽管庞加莱经常谈到确定性宇宙，但他说潜意识中新可能性的产生涉及到[随机性|机会]]。

可以肯定的是，在经过一段长时间的无意识工作之后，以一种突然的光明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我自动作用的结果，但是那些有趣的组合却进入了意识领域。。。只有少数人是和谐的，因此同时又是有用的和美丽的，它们将能够影响我所说的几何学家的特殊情感；一旦被唤起，就会把我们的注意力引向它们，从而使它们有机会变得有意识。。。与此相反，在潜意识自我中，存在着我称之为自由的统治，如果一个人可以把这个名字命名为纯粹的缺乏纪律和偶然产生的混乱。<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>

-----
==Bibliography参考文献==

Other:

其他:

===Poincaré's writings in English translation庞加莱的英语翻译作品===

Popular writings on the [[philosophy of science]]:
关于[[科学哲学]]的通俗著作：

*{{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.
*1904年“科学与假设”，沃尔特·斯科特出版公司。

* 1913. "The New Mechanics," The Monist, Vol. XXIII.
*1913年。”《新力学》，《一元论》，第二十三卷。
*1913. "The Relativity of Space," The Monist, Vol. XXIII.
*1913年。”空间的相对性，《一元论》，第二十三卷。
*1913. {{Citation | title=Last Essays. |place=New York |publisher=Dover reprint, 1963 | url=https://archive.org/details/mathematicsandsc001861mbp}}
*1913年。{{引文|标题=最后一篇论文。|place=New York | publisher=Dover再版，1963 |网址=https://archive.org/details/mathematicsandsc001861mbp}}
*1956. ''Chance.'' In James R. Newman, ed., The World of Mathematics (4 Vols).
*1956年《机会》，詹姆斯·R·纽曼主编，《数学世界》（4卷）。
*1958. ''The Value of Science,'' New York: Dover.
*1958年《科学的价值》，纽约：多佛。

On [[algebraic topology]]:
关于[[代数拓扑]]：

* 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]]:
关于[[天体力学]]：

* 1892–99. ''New Methods of Celestial Mechanics'', 3 vols. English trans., 1967. {{isbn|1-56396-117-2}}.
*1892-1899年。”天体力学新方法》，3卷。英语译本，1967年。{{isbn | 1-56396-117-2}}。
* 1905. "The Capture Hypothesis of J. J. See," The Monist, Vol. XV.
*1905年。”J.J.的捕获假说，见《一元论》，第十五卷。
* 1905–10. ''Lessons of Celestial Mechanics''.
*1905-10年。”天体力学课程”。

On the [[philosophy of mathematics]]:
关于[[数学哲学]]：
* 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:

*伊瓦尔德，威廉B.，编辑，1996年从康德到希尔伯特：数学基础中的一本原著，2卷。牛津大学出版社。包含庞加莱的以下作品：
** 1894, "On the Nature of Mathematical Reasoning," 972–81.
**1894年，“论数学推理的本质”，972-81。
** 1898, "On the Foundations of Geometry," 982–1011.
**1898年，《几何基础》，982-1011年。
** 1900, "Intuition and Logic in Mathematics," 1012–20.
**1900年，“数学中的直觉和逻辑”，1012-20。
** 1905–06, "Mathematics and Logic, I–III," 1021–70.
**1905年至1906年，“数学与逻辑，I–III”，1021–70。
** 1910, "On Transfinite Numbers," 1071–74.
**1910年，“关于超限数”，1071-74。
*1905. "The Principles of Mathematical Physics," The Monist, Vol. XV.
*1905年。”《数学物理原理》，《一元论》，第十五卷。
* 1910. "The Future of Mathematics," The Monist, Vol. XX.
*1910年。”《数学的未来》，《一元论》，第二十卷。
*1910. "Mathematical Creation," The Monist, Vol. XX.
*1910年。”《数学创造》，《一元论》，第二十卷。

Other:
其他：
* 1904. ''Maxwell's Theory and Wireless Telegraphy,'' New York, McGraw Publishing Company.
*1904年《麦克斯韦理论与无线电报》，纽约，麦格劳出版公司。
* 1905. "The New Logics," The Monist, Vol. XV.
*1905年。”《新逻辑学》，《一元论》，第十五卷。
* 1905. "The Latest Efforts of the Logisticians," The Monist, Vol. XV.
*1905年。”《后勤人员的最新努力》，《一元论》，第十五卷。

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请参阅==

===Concepts概念===

* [[Poincaré complex]] – an abstraction of the singular chain complex of a closed, orientable manifold
*[[Poincarécomplex]]——封闭可定向流形的奇异链复形的抽象
* [[Poincaré duality]]
*[[庞加莱双重性]]
* [[Poincaré disk model]]
*[[庞加莱磁盘模型]]
* [[Poincaré group]]
*[[庞加莱集团]]

}}

}}

* [[Poincaré half-plane model]]
*[[庞加莱半平面模型]]
* [[Homology sphere#Poincaré homology sphere|Poincaré homology sphere]]
*[[同调球#庞加莱同调球|庞加莱同调球]]
*[[Poincaré inequality]]
*[[庞加莱不等式]]
*[[Poincaré map]]
*[[Poincarémap]]

* [[Poincaré residue]]

*[[庞加莱残留物]]
* [[Poincaré series (modular form)]]
*[[庞加莱系列（模块形式）]]
* [[Poincaré space]]
*[[庞加莱空间]]
* [[Poincaré metric]]
* [[Poincaré plot]]

* [[Hilbert–Poincaré series|Poincaré series]]
*[[希尔伯特-庞加莱系列|庞加莱系列]]
* [[Poincaré sphere (optics)|Poincaré sphere]]
*[Poincarésphere（光学）| Poincarésphere]]
* [[Poincaré–Lelong equation]]
*[Poincaré–Lelong方程]]
* [[Poincaré–Lindstedt method]]
*[Poincaré–Lindstedt方法]]
* [[Poincaré–Lindstedt perturbation theory]]
*庞加莱-林德斯特摄动理论
*[[Poincaré–Steklov operator]]
*[[Poincaré–Steklov运算符]]
* [[Reflecting Function]]
*[[反射函数]]

===Theorems理论===

|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.
*[[庞加莱递推定理]]：某些系统在足够长但有限的时间后，将返回到非常接近初始状态的状态。

|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.
*[[Poincaré–Bendixson定理]：关于平面、圆柱或两个球体上连续动力系统轨道的长期行为的陈述。

|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–Hopf定理]]：毛球定理的一个推广，它指出在没有源或汇的球体上没有光滑的向量场。

* [[Poincaré–Lefschetz duality theorem]]: a version of Poincaré duality in geometric topology, applying to a manifold with boundary
*[[Poincaré–Lefschetz对偶定理]]：几何拓扑中Poincaré对偶的一个版本，适用于有边界的流形

* [[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.
*[[庞加莱分离定理]]：给出了实对称矩阵B'AB的特征值的上下界，它可以看作是更大的实对称矩阵a在B列所跨的线性子空间上的正交投影。

* [[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定理]]：沿相反方向旋转两个边界的环空间的每个保面积、保方向同胚至少有两个不动点。

* [[Poincaré–Birkhoff–Witt theorem]]: an explicit description of the universal enveloping algebra of a Lie algebra.
*[[Poincaré–Birkhoff–Witt定理]]：李代数的泛包络代数的显式描述。

* [[Poincaré conjecture]] (now a theorem): Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
*[[Poincaré猜想]]（现在是一个定理）：每个单连通的闭3-流形都同胚于3-球面。

* [[Poincaré–Miranda theorem]]: a generalization of the [[intermediate value theorem]] to ''n'' dimensions.
*[[Poincaré–Miranda定理]]：将[[中值定理]]推广到“n”维。

=== Other其他 ===

{{Columns-list|colwidth=30em|

*[[French epistemology]]
*[[法国认识论]]
*[[History of special relativity]]
*[[狭义相对论史]]

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

*[[以亨利·彭加勒命名的事物列表]]
*[[Institut Henri Poincaré]], Paris

*[[Henri Poincaré学院]]，巴黎
*[[Brouwer fixed-point theorem]]
*[[Brouwer不动点定理]]

*[[Relativity priority dispute]]

*[[相对论优先权争议]]

*[[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>

*[[认知结构现实主义]]<ref>[http://plato.stanford.edu/entries/structural reality/#Rel“结构现实主义”]：詹姆斯·拉迪曼在“[[斯坦福哲学百科全书]]”中的词条</ref>

}}

==References参考==

===Footnotes脚注===

{{Reflist}}

===Sources资源===

* [[Eric Temple Bell|Bell, Eric Temple]], 1986. ''Men of Mathematics'' (reissue edition). Touchstone Books. {{isbn|0-671-62818-6}}.
*[[Eric Temple Bell | Bell，Eric Temple]]，1986年《数学人》（再版）。试金石书籍。{{isbn | 0-671-62818-6}}。

* Belliver, André, 1956. ''Henri Poincaré ou la vocation souveraine''. Paris: Gallimard.
*安德烈·贝利弗，1956年亨利·彭加勒·奥拉·苏维拉尼职业”。巴黎：加利玛。
*[[Peter L. Bernstein|Bernstein, Peter L]], 1996. "Against the Gods: A Remarkable Story of Risk". (p. 199–200). John Wiley & Sons.
*[[Peter L.Bernstein | Bernstein，Peter L]]，1996年。”反对上帝：一个关于风险的非凡故事”。（第199-200页）。约翰威利父子公司。
*[[Carl Benjamin Boyer|Boyer, B. Carl]], 1968. ''A History of Mathematics: Henri Poincaré'', John Wiley & Sons.
*[[Carl Benjamin Boyer | Boyer，B.Carl]]，1968年数学史：亨利庞加莱，约翰威利和儿子。
*[[IvorGrattan-Guinness|Grattan-Guinness, Ivor]], 2000. ''The Search for Mathematical Roots 1870–1940.'' Princeton Uni. Press.
*[Ivor Grattan吉尼斯| Grattan Guinness，Ivor]]，2000年1870-1940年数学根源的探索。按。
* {{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.
*{{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}}.网络版发表在2004年美国医学会杂志上。

* Folina, Janet, 1992. ''Poincaré and the Philosophy of Mathematics.'' Macmillan, New York.
*福琳娜，珍妮特，1992年彭加勒与数学哲学〉，麦克米伦，纽约。
* [[Jeremy Gray|Gray, Jeremy]], 1986. ''Linear differential equations and group theory from Riemann to Poincaré'', Birkhauser {{isbn|0-8176-3318-9}}
*1986年从黎曼到庞加莱的线性微分方程与群论，Birkhauser{isbn | 0-8176-3318-9}}
*Gray, Jeremy, 2013. ''Henri Poincaré: A scientific biography''. Princeton University Press {{isbn|978-0-691-15271-4}}
*格雷，杰里米，2013年彭加勒：科学传记”。普林斯顿大学出版社
*{{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 }}
*{{引文|网址=http://www.ams.org/notices/200509/comm-mawhin.pdf

|标题=亨利·彭加勒。为科学服务的一生

|作者=Jean Mawhin | journal=AMS通知

|日期=2005年10月|卷=52 |期=9 |页=1036–1044}

*Kolak, Daniel, 2001. ''Lovers of Wisdom'', 2nd ed. Wadsworth.
*科拉克，丹尼尔，2001年《智慧的情人》，第二版，沃兹华斯。

*Gargani, Julien, 2012. ''Poincaré, le hasard et l'étude des systèmes complexes'', L'Harmattan.
*加加尼，朱利安，2012年庞加莱，《哈萨德与系统综合体》，哈马丹。

* Murzi, 1998. "Henri Poincaré".
*穆尔齐，1998年。”亨利·彭加勒”。
*O'Connor, J. John, and Robertson, F. Edmund, 2002, "Jules Henri Poincaré". University of St. Andrews, Scotland.
*欧康纳，J.约翰和罗伯逊，F.埃德蒙，2002年，“朱尔斯-亨利-庞加莱”。苏格兰圣安德鲁斯大学。
*[[Ivars Peterson|Peterson, Ivars]], 1995. ''Newton's Clock: Chaos in the Solar System'' (reissue edition). W H Freeman & Co. {{isbn|0-7167-2724-2}}.
*[[Ivars Peterson | Peterson，Ivars]]，1995年牛顿钟：太阳系的混沌》（再版）。W H弗里曼公司{isbn | 0-7167-2724-2}。

*Sageret, Jules, 1911. ''Henri Poincaré''. Paris: Mercure de France.
*萨盖特，朱尔斯，1911年亨利·彭加勒”。巴黎：法国美爵酒店。
* Toulouse, E.,1910. ''Henri Poincaré''.—(Source biography in French) at University of Michigan Historic Math Collection.
*图卢兹，E.，1910年亨利·彭加勒（法国传记来源）密歇根大学历史数学收藏。
*{{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}}
*{引用图书| title=Mathematics and Its History | edition=3rd，插图| first1=John | last1=Stillwell | publisher=Springer Science&Business Media |年份=2010 | isbn=978-1-4419-6052-8 |网址=https://books.google.com/books？id=V7mxZqjs5yUC | ref=harv}}
*[[F. Verhulst|Verhulst, Ferdinand]], 2012 ''Henri Poincaré. Impatient Genius''. N.Y.: Springer.
*[[F.Verhulst | Verhulst，Ferdinand]]，2012年，“亨利·彭加勒。“不耐烦的天才”。N、 Y：斯普林格。

*''Henri Poincaré, l'œuvre scientifique, l'œuvre philosophique'', by Vito Volterra, Jacques Hadamard, Paul Langevin and Pierre Boutroux, Felix Alcan, 1914.
*保罗·阿苏瓦·阿库维·保罗·阿库维·费利瓦尼·波依瓦，1914年。
** ''Henri Poincaré, l'œuvre mathématique'', by [[Vito Volterra]].
**“亨利·彭加勒，l’uvre matique”，作者[[Vito Volterra]]。
** ''Henri Poincaré, le problème des trois corps'', by [[Jacques Hadamard]].
**“亨利·彭加勒，特洛伊军团的问题”，作者[[Jacques Hadamard]]。
** ''Henri Poincaré, le physicien'', by [[Paul Langevin]].
**“亨利·彭加勒，物理学”，作者[[Paul Langevin]]。
** ''Henri Poincaré, l'œuvre philosophique'', by [[Pierre Boutroux]].
**“亨利·彭加勒，l’uvre Phologique”，作者[[Pierre Boutroux]]。
*{{PlanetMath attribution|id=3793|title=Jules Henri Poincaré}}
*{{PlanetMath属性| id=3793 | title=Jules-Henri-Poincaré}}

==Further reading延伸阅读==

===Secondary sources to work on relativity是研究相对论的第二资源===

* {{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非主流来源===

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外部链接==

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 – 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

{{s-start}}

Category:Thermodynamicists

类别: 热力学家

{{s-culture}}

Category:Fluid dynamicists

类别: 流体动力学家

{{s-bef|before=[[Sully Prudhomme]]}}

Category:Topologists

类别: 拓扑学家

{{s-ttl|title=[[List of members of the Académie française#Seat 24|Seat 24]] [[Académie française]] 1908–1912|years=}}

Category:University of Paris faculty

类别: 巴黎大学教员

{{s-aft|after=[[Alfred Capus]]}}

Category:French male writers

类别: 法国男性作家

{{s-end}}

Category:Deaths from embolism

类别: 死于栓塞

{{philosophy of science}}

Category:Dynamical systems theorists

范畴: 动力系统理论家

<noinclude>

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

[[Category:待整理页面]]

亨利·庞加莱 Jules Henri Poincaré

2021-05-30T09:23:01Z

INCH RONG：

此词条暂由水流心不竞初译，翻译字数共，未经审校，带来阅读不便，请见谅。

{{#seo:
|keywords=亨利·庞加莱 Henri Poincaré，复杂性理论，复杂性经济学，自主经济学
|description=庞加莱猜想，三体问题，拓扑学，狭义相对论
}}
== 基本信息 ==
[[File:Henri_Poincare.jpg|230px|left]]

{| class="wikitable" width="50%"
|-
! 类别 !! 信息
|-
| 姓名 || 布莱恩·阿瑟 W. Brian Arthur
|-
| 出生日期 || 1945年7月31日
|-
| 出生地 || Belfast, Northern Ireland 北爱尔兰贝尔法斯特
|-
|居住地 || Northern California 北加利福尼亚
|-
| 所在机构 || 斯坦福大学、圣达菲研究所、帕罗奥多研究中心系统科学实验室
|-
| 主要研究方向 || 复杂性理论、[[复杂经济学]]和[[自主经济学]]等领域
|-
| 主要贡献 || 埃尔法罗尔酒吧问题 [[El Farol Bar problem]]、收益递增理论 [[Increasing returns]]等

|}

 

== 个人介绍 ==
布莱恩·阿瑟 W. Brian Arthur是著名经济学家，圣塔菲研究所外聘教授，帕罗奥多研究中心系统科学实验室访问研究员。阿瑟在上世纪八十年代，受诺贝尔经济学奖得主肯尼斯·阿罗的邀请，来到刚刚建立的圣塔菲研究所。在圣塔菲研究所，阿瑟将复杂系统理论与经济学研究结合在一起，逐渐形成了“[[复杂经济学]]”的构想，创立了对经济学和复杂系统的跨学科研究新模式。
作为圣塔菲研究所最早一批研究复杂性的学者，阿瑟是复杂性科学领域的奠基人之一，由于其突出成绩，于2008年荣获复杂性科学领域首届拉格朗日奖。阿瑟提出了关于经济学中[[收益递增]]的概念，尤其放大经济中的小型随机事件的影响，这项工作也成为了理解高新技术经济的基础。凭借[[收益递增]]理论，阿瑟于1990年获得熊彼特奖。

== 教育经历 ==
* 1973年，获得加州大学伯克利分校运筹学博士学位，博士辅修专业：金融
* 1973年，获得加州大学伯克利分校经济学硕士学位
* 1969年，获得密歇根大学数学硕士学位
* 1967年，获得英国兰卡斯特大学运筹学硕士学位
* 1966年，获得贝尔法斯特皇后大学电气工程学士学位

== 任职经历 ==
* 1994年至今圣达菲研究所花旗银行教授
* 1998-2000年普华永道研究员
* 1983-1996年斯坦福大学经济学和人口研究所院长和弗吉尼亚·莫里森教授
* 1988年至今圣达菲研究所科学委员会成员，1987年起；董事会成员，1994年起；外聘教授，1998-1994年；1988-1989年和1995年指导SFI的第一个研究项目(经济学)
* 1977-1982年系统和决策科学组研究学者；奥地利拉克森堡国际应用系统分析研究所学者
* 1974-1977年纽约人口理事会助理

== 学术背景 ==
=== 研究方向 ===
1、复杂性理论

2、技术如何演变

3、复杂经济学

4、自主经济学

=== 合作学者 ===
* [https://scholar.google.com/citations?user=CiwEc6gAAAAJ&hl=en Blake LeBaron]
* [https://scholar.google.com/citations?user=91RtF7oAAAAJ&hl=en Wolfgang Polak]
* [https://scholar.google.com/citations?user=O5SjWqAAAAAJ&hl=en Andrzej Ruszczyński]

=== 代表性文章 ===
# "Complexity Economics: A Different Framework for Economic Thought," in The Economy and Complexity, Oxford U Press, 2014.
# "All Systems will be Gamed: Exploitive Behavior in Policy Systems," in The Economy and Complexity, Oxford U Press, 2014.
# "Do Markets Lead Always to Best Outcomes? A Comment on Neil Kay's paper," Research Policy, 2013.
# "The Second Economy," McKinsey Quarterly, Oct-Nov 2011.
# "The Structure of Invention," Research Policy, 36,2, March 2007.
# "The Evolution of Technology in a Simple Computer Model," with W. Polak, Complexity, 11, 5, May 2006.
# "Agent-Based Modeling and Out-Of-Equilibrium Economics," Handbook of Computational Economics, Vol. 2, K. Judd and L. Tesfatsion, eds, Elsevier/North-Holland, 2006.
# "Cognition: the Black Box of Economics," in The Complexity Vision and the Teaching of Economics, D. Colander, ed., 2000, Edward Elgar Publishers.
# "Time Series Properties of an Artificial Stock Market," with B. LeBaron and R. Palmer, Journal of Economic Dynamics and Control, 23, 1487-1516, 1999.
# "Complexity and the Economy," Science, 2 April 1999, 284, 107-109. Reprinted in The Complexity Vision and the Teaching of Economics, D. Colander, ed., Edward Elgar Publishers, 2000.
# "Asset Pricing Under Endogenous Expectations in an Artificial Stock Market," with J.H. Holland, B. LeBaron, R. Palmer, and P. Tayler, SFI Paper 96-12-093, Economic Notes. Reprinted in The Economy as an Evolving Complex System II. Edited (with S. Durlauf and D. Lane), Addison-Wesley, 1997.
# "Beyond Rational Expectations: Indeterminacy in Economic and Financial Markets" in Frontiers of the New Institutional Economics, J.N. Drobak and J.V. Nye (eds.), Academic Press, 1997.

== 荣誉成就 ==
* 2009年获得兰开斯特大学荣誉科学博士学位
* 2008年获得首届复杂性科学拉格朗日奖
* 2000年获得爱尔兰国立大学荣誉经济科学博士学位
* 1990年获得熊彼特经济学奖
* 1987年被授予古根海姆奖学金

== 联系方式 ==
* 地址：Systems Sciences Lab, PARC 3333 Coyote Hill Rd, Palo Alto,CA 94304 USA
* 电子邮箱：mprandoni@gmail.com
* 办公电话：+1 505 989 1014

== 其他信息 ==
* [http://tuvalu.santafe.edu/~wbarthur/Bio_Info/Resume_Web.pdf PDF版简历，更新到2015年]
* [https://en.wikipedia.org/wiki/W._Brian_Arthur 维基百科介绍]
* [http://www.santafe.edu/arthur 布莱恩·阿瑟的个人主页]

== 与集智的关系==
[[File: 布莱恩·阿瑟.jpg |400px|thumb|布莱恩·阿瑟在集智俱乐部的演讲]]
[https://swarma.org/?p=17050 阿瑟受集智俱乐部邀请于2019年9月20日在上海举办《科技把经济带往何方》的主题讲座 | AI&Society系列活动第18期]

== 相关报道 ==
=== 视频内容 ===
[https://www.youtube.com/watch?v=P8IzaECeQOk W. Brian Arthur spoke to the Stanford Complexity Group on Dec. 3, 2015 布莱恩·阿瑟在2015年12月3日斯坦福复杂性小组发表的讲话]

=== 文章报道 ===
[https://www.strategy-business.com/article/16402?gko=d4710 An Interview with W. Brian Arthur 关于布莱恩·阿瑟访谈]

[https://www.fastcompany.com/3064681/most-important-economic-theory-in-technology-brian-arthur A Short History Of The Most Important Economic Theory In Tech 技术领域最重要的经济理论简史]

== 最新事件 ==
* [https://a16z.com/2018/05/16/network-effects-positive-feedbacks-increasing-returns-complexity-silicon-valley-history-innovation/ 阿瑟与马克·安德森在博客上讨论网络效应、硅谷和人工智能等问题]
* [https://www.mckinsey.com/business-functions/mckinsey-analytics/our-insights/Where-is-technology-taking-the-economy 在麦肯锡季刊上发布一篇新文章：《技术将经济带向何处？》]
* [https://www.mckinsey.com/business-functions/mckinsey-analytics/our-insights/Where-is-technology-taking-the-economy 在Fast公司发布一篇文章：《技术中最重要理论的简史》]

== 相关推荐 ==
=== 书籍推荐 ===
====[https://www.amazon.com/Complexity-Economy-W-Brian-Arthur/dp/0199334293 《复杂经济学》--《Complexity and the Economy》]====

[[File: 复杂经济学.jpeg|200px|thumb|right|复杂经济学]]
1999年，阿瑟在《科学》杂志上发表的重要论文引入了复杂性和经济的概念。自阿瑟第一次提出这一概念以来的15年里，他一直在探索经济进化的复杂性，使用创新和不寻常的理论技术，提炼他早期的观点，并提出新的方法来概念化经济理论的基础。这本书里的文章都是这种恰当探索的结果。这本书是一个丰富的思想来源，所有的结构都在彻底创新的框架内，呈现出一个大师级散文作家的清晰，强烈推荐。--K. V. Veupillai, CHOICE Magazine

“阿瑟的书表明，有许多新的方法和技术可以用来改变我们对经济的看法，并提出政策和法律建议。”-- Andrew Sheng

● 经典的、强调“均衡”的经济学理论建构的是一个充满了确定性的理想世界，在这个世界里，每个人只要按照标准的演绎推理和标准的均衡分析，就能找到最好的决策。而复杂经济学则向我们描绘了一个充满不确定的世界。在这个世界里，有大量的创新在不断涌现，我们每个人都要对这个世界充满开放的心态，接受这个世界的变动和改变，适应这些变化，调整自己的行为。

● 复杂经济学是相对传统经济学提出的全新概念。它从最根本处挑战传统经济学：传统经济学认为人是自私与理性的，而且认为“均衡”是经济常态。与之相对，复杂经济学认为人是非理性的，而且“非均衡”才是经济常态，“均衡”反倒是经济的特例。在颠覆最根本的前提假设下，复杂经济学从宏观上论述何为之“复杂”——多元互动，持续变化的。在此基础上提出了与传统经济学“边际效益递减”规律相反的“收益递增”的正反馈现象。

[[File: 技术的本质.jpg |150px|thumb|right|技术的本质]]
====[https://www.amazon.com/Nature-Technology-What-How-Evolves/dp/1416544062 《技术的本质》--《The Nature of Technology: What It Is and How It Evolves》]====

这本书解释了变革性新技术是如何产生的，以及创新是如何工作的。传统思维将技术的发明归因于“跳出框架思考”，或者模糊地归因于天才或创造力，但阿瑟指出，这种解释是不充分的。发明是解决问题——满足需求——将现有的部分组合在一起，并解决和重新解决一路上出现的问题。

● 本书是打开"技术黑箱"的钥匙，它用平实的语言将技术本质的思想娓娓道来。

● 构建了关于技术的理论体系，阐明了技术的本质及其进化机制。

● 技术思想领域的开创性作品。

[[File:经济中的收益递增和路径依赖.jpg |150px|thumb|right|经济中的收益递增和路径依赖]]
====[https://www.amazon.com/dp/0472064967/ref=cm_sw_su_dp 《经济中的收益递增和路径依赖》--《Increasing Returns and Path Dependence in the Economy》]====

这是阿瑟在1982年至1992年间撰写的关于收益递增的论文集。大部分文章探讨了在不断增加的回报下的市场动态，特别是正反馈在锁定单一主导产品、技术或公司中的作用。并且介绍或讨论了许多对技术经济至关重要的概念:锁定、路径依赖、网络外部性、小历史事件的作用、正反馈、竞争技术和赢家通吃等。

=== 集智课程推荐 ===
====[https://www.zaojiu.com/lives/5d761b97ddc8f60001c52210 科技把经济带往何方]====
2019年9月20日莱恩·阿瑟（W. Brian Arthur）在集智俱乐部进行的以《科技把经济带往何方》为主题的演讲视频，讨论了对下列问题的看法：
* 人工智能等技术变革，会如何影响经济系统的分配规则
* 数字经济（虚拟经济）与实体经济的关系是怎样的
* 数字经济的变革，如何影响经济系统的发展，进而改变社会结构
* 技术变革带对经济的影响，是在持久深入还是日渐平缓
* 如何在技术创新在带来社会福利的同时，消减随之而来的隐私、分配等社会问题
* 当工作机会被新技术挤占，人类如何找到生活的意义

 

=== 文章推荐 ===
====[https://swarma.org/?p=16621 布莱恩·阿瑟：数字经济是增长新引擎，还是潘多拉魔盒？]====
本文译自著名经济学家布莱恩·阿瑟2011年发表的经典文章The second economy，阿瑟在文中发表了对数字经济的一系列论述和预言。

-----
本条目由[[用户:峰峰|峰峰]]编辑，如有问题，欢迎在讨论页面进行讨论。

'''本词条内容源自公开资料，遵守 CC3.0协议。'''

[[Category:复杂经济学]]
[[Category:人物]]
ered 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

<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>

<blockquote>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>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<math>\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>\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

并证明了任意函数<math>\ell\left（\varepsilon\right）</math>对于所有<math>\varepsilon</math>必须是统一的（Lorentz通过一个不同的参数设置<math>\ell=1</math>），以使变换形成一个组。在1906年发表的论文的放大版中，庞加莱指出组合<math>x^2+y^2+z^2-c^2t^2</math>是[[不变量（数学）|不变量]]。他指出，通过引入<math>ct\sqrt{-1}</math>作为第四个虚坐标，Lorentz变换仅仅是四维空间中绕原点的旋转，他使用了[[四向量]]s的早期形式。<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 = 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.

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Hertzian oscillato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.

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。<ref>Walter (2007), Secondary sources on relativity</ref>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 [[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''2. 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[[质量-能量等效性#电磁静止质量|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[[牛顿运动定律|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。<ref name=action/>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''/''c''2。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).<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:

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。

{{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. }}

{{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>\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]].

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>\gamma m</math>所暗示的质量非守恒性，亚伯拉罕的变质量理论和[[瓦尔特·考夫曼（物理学家）| 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。

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

<center>环面向杯子的拓扑变换</center>

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''/''c''2 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>

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''/''c''2<ref name=darrigol>Darrigol 2005, Secondary sources on relativity</ref> 解决了庞加莱悖论，没有使用以太内部的任何补偿机制。<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}}请参阅 [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.<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."''

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”）。<ref name=“1905 paper”/>“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,<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>

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。<ref name=long/>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。<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}}另见本函及其评注{{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> 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……”<ref>Darrigol 2004, Secondary sources on relativity</ref>

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

<center> 三体问题的混乱运动(计算机模拟) </center>

====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。

{{Further|History of special relativity|Relativity priority dispute}}
{{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,<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>

庞加莱在狭义相对论的发展中的工作是公认的，<ref name=darrigol />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。<ref>Galison 2003 and Kragh 1999, Secondary sources on relativity</ref> 。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.<ref>Whittaker 1953, Secondary sources on relativity</ref>

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

===Algebra and number theory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]].<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.
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。<ref>Poincaré, Selected works in three volumes. page = 682{{full citation needed|date=September 2019}}</ref> 他还对离散群理论及其表示法做出了重大贡献。

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

[[图片：马克杯和环面变形.gif|右| frame | 50px |<center>圆环到杯子的拓扑变换</center>]]

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

<gallery caption="积分曲线的奇点" >

File: Phase Portrait Sadle.svg | Saddle

文件: Phase Portrait Sadle.svg | Saddle

===Topology拓扑学===

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}}

[[费利克斯·克莱因]]在他的“Erlangen程序”（1872）中明确地定义了这个主题：任意连续变换的几何不变量，一种几何学。正如[[约翰本尼迪克特名单]]所建议的，引入了术语“拓扑”，而不是以前使用的“分析位置”。一些重要的概念是由[[恩里科·贝蒂]]和[[伯恩哈德·黎曼]]介绍的。但这一科学的基础，对于任何维度的空间，都是由庞卡莱创造的。他关于这个主题的第一篇文章发表在1894年{{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>

他对几何学的研究导致了[[同伦]]和[[同伦（数学）|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，如贝蒂Betti数和[[基本群]]。Poincaré证明了n维多面体的边数、顶点数和面数的公式（Euler-Poincaré定理），给出了维数直观概念的第一个精确表达式。<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天文学与天体力学===

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

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>

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。<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.

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).<ref>A. Kozenko, The theory of planetary figures, pages = 25–26{{full citation needed|date=September 2019}}</ref>

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。<ref>A. Kozenko, The theory of planetary figures, pages = 25–26{{full citation needed|date=September 2019}}</ref>

===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).<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>

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。<ref>French: "Mémoire sur les courbes définies par une équation différentielle"</ref>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。<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性格==

[[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.

庞加莱的工作习惯被比作蜜蜂从一朵花飞到另一朵花。庞加莱对他的思维方式很感兴趣；他研究了自己的习惯，并于1908年在巴黎普通心理学研究所发表了一篇关于他的观察结果的演讲。他把自己的思维方式与他如何取得几个发现联系在一起。

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  believed that logic was not a way to invent but a way to structure ideas and that logic limits ideas.)

数学家达布克斯声称他是“不直觉的”（直觉的），认为这一点可以从他经常通过视觉表现来工作的事实中得到证明。他不在乎严谨，也不喜欢逻辑。<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> （尽管如此，[[Jacques Hadamard]]写道，庞加莱的研究显示出惊人的清晰性。<ref>J. Hadamard. L'oeuvre de H. Poincaré. Acta Mathematica, 38 (1921), p. 208</ref>庞加莱自己也写道他<！--TODO:加上庞加莱对严谨的看法，参见http://www.forgottenbooks.org/readbook/American_Journal_of_Mathematics_1890_v12_1000084889#233 —每次我能做到的时候，我都是绝对严谨的-->相信逻辑不是一种发明的方式，而是一种构建思想的方式，逻辑限制了思想。）

===Toulouse's characterisation图卢兹的特征化===

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:

庞加莱的心理组织不仅令庞加莱本人感兴趣，而且巴黎高等研究院心理学实验室的心理学家埃杜阿尔德·图卢兹也感兴趣。图卢兹写了一本书，名叫《亨利·庞加莱》（1910年）<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>在其中，他讨论了庞加莱的常规日程安排：

* 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.
*他每天在同一时间短时间内工作。他每天进行四个小时的数学研究，从上午10点到中午，然后再从下午5点到晚上7点。。晚上晚些时候他会在杂志上读文章。

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.
*他通常的工作习惯是在脑子里彻底解决一个问题，然后把完成的问题写在纸上。

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.
*他两手灵巧，近视。

* 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.
*当他去听课时，他对所听到的东西进行视觉化的能力被证明是特别有用的，因为他的视力很差，他不能正确地看到讲课者在黑板上写的东西。

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.

这些能力在某种程度上被他的缺点所抵消：

*他身体笨拙，艺术上拙劣。

*他总是匆匆忙忙的，不喜欢回去修改或更正。

*他从来没有在一个问题上花很长时间，因为他相信潜意识会在他有意识地处理另一个问题时继续工作。

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，图卢兹指出，大多数数学家都是从已经确立的原则出发，而庞加莱每次都是从基本原理出发（O'Connor et al.，2002）。

His method of thinking is well summarised as:

他的思维方法概括如下：

{{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)}}

{{quote|text=“他习惯于忽略细节，只看山顶，以惊人的速度从一个瞬间到另一个瞬间，发现的事实围绕着他们的中心聚集在一起，并自动地被归入他的记忆中。”|sign=Belliver (1956)}}

Named after him

以他的名字命名

===Attitude towards transfinite numbers对超限数的态度===

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>
庞加莱对[[Georg Cantor]]的[[超限数]]理论感到沮丧，并将其称为数学最终将被治愈的“疾病”。<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

庞加莱说：“没有真正的无限；坎托利亚人忘记了这一点，这就是他们陷入矛盾的原因。”<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荣誉==

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'''
“奖项”
<blockquote>

< 封锁报价 >

*Oscar II, King of Sweden's mathematical competition (1887)
*奥斯卡二世，瑞典数学竞赛之王（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>
*[[荷兰皇家艺术和科学院]外籍院士（1897年）<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
*[[美国哲学学会]]1899年
*[[Gold Medal of the Royal Astronomical Society]] of London (1900)
*伦敦（1900年）的[[皇家天文学会金牌]]

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
*1905 [博利奖]
*[[Matteucci Medal]] 1905
* [ [马图基奖章] ] 1905
*[[French Academy of Sciences]] 1906
*[[法国科学院]]1906年
*[[Académie française]] 1909
*〔〔阿克米-弗兰〕〕1909
*[[Bruce Medal]] (1911)
＊[〔布鲁斯奖章〕〕（1911）

Popular writings on the philosophy of science:

关于科学哲学的通俗著作:

'''Named after him'''
以他命名的
|author=Poincaré, Henri

| author = poincaré，Henri

*[[Institut Henri Poincaré]] (mathematics and theoretical physics center)
*[[亨利庞加莱学院]]（数学与理论物理中心）

|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
*月球上的陨石坑 [[庞加莱（陨石坑)|庞加莱]]

|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]]
*小行星
*[[List of things named after Henri Poincaré]]
*[[以亨利·庞加莱命名的事物列表]]

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哲学==

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]]'':

庞加莱的哲学观点与[[伯特兰·罗素]]和[[哥特罗布·弗雷格]]相反，他们认为数学是[[逻辑]]的一个分支。庞加莱强烈反对，声称[[直觉（知识）|直觉]]是数学的生命。庞加莱在他的书“[[科学与假设]]”中提出了一个有趣的观点：

| 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.}}

{{quote | text=对于一个肤浅的观察者来说，科学的真理是不容置疑的；科学的逻辑是绝对正确的，如果科学家有时是错误的，这仅仅是因为他们错误地理解了它的规则}}

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}}.

庞加莱认为[[算术]]是[[分析/合成区别|合成]]。他认为[[皮亚诺的公理]]不能用归纳法原理进行非循环证明（Murzi，1998），因此得出结论认为算术是“综合的而非分析的”。庞加莱接着说，数学不能从逻辑中推导出来，因为它不是分析性的。他的观点与[[Immanuel Kant]]的观点相似（Kolak，2001，Folina 1992）。他强烈反对Cantorian[[set theory]]，反对其使用[[不精确性|非指示性]]定义{{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>

然而，庞加莱并没有在哲学和数学的所有分支中分享康德的观点。例如，在几何学中，庞加莱认为[[非欧几里德几何|非欧几里德空间]]的结构可以通过分析得到。庞加莱认为传统在物理学中起着重要的作用。他的观点（以及后来一些更极端的版本）被称为“[[传统主义]]”。<ref>Yemima Ben-Menahem, ''Conventionalism: From Poincare to Quine'', Cambridge University Press, 2006, p. 39.</ref>庞加莱认为[[牛顿第一定律]]不是经验性的，而是力学的传统框架假设（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>他还认为物理空间的几何学是传统的。他考虑了物理场的几何结构或温度梯度可以改变的例子，要么将一个空间描述为由刚性标尺测量的非欧几里德空间，要么描述为标尺通过可变热分布而膨胀或收缩的欧几里德空间。然而，庞加莱认为我们太习惯了[[欧几里德几何]]，我们宁愿改变物理定律来保存欧几里德几何，而不是转向非欧几里德物理几何。<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自由意志===

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>

庞加莱在巴黎心理学学会之前的著名演讲（出版为“[[科学与假设]]”、[[科学的价值]]”和“科学与方法”）被[[Jacques Hadamard]]引用为创意和发明由两个心理阶段组成的思想来源，首先是可能的解决方案的随机组合一个问题，然后是一个批判性的评估<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]].
尽管庞加莱经常谈到确定性宇宙，但他说潜意识中新可能性的产生涉及到[随机性|机会]]。

<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>

可以肯定的是，在经过一段长时间的无意识工作之后，以一种突然的光明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我自动作用的结果，但是那些有趣的组合却进入了意识领域。。。只有少数人是和谐的，因此同时又是有用的和美丽的，它们将能够影响我所说的几何学家的特殊情感；一旦被唤起，就会把我们的注意力引向它们，从而使它们有机会变得有意识。。。与此相反，在潜意识自我中，存在着我称之为自由的统治，如果一个人可以把这个名字命名为纯粹的缺乏纪律和偶然产生的混乱。<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>

庞加莱的两阶段随机组合和选择成为[[Daniel Dennett]]自由意志两阶段模型的基础。<ref>Dennett, Daniel C. 1978. Brainstorms: Philosophical Essays on Mind and Psychology. The MIT Press, p.293</ref>

==Bibliography参考文献==

Other:

其他:

===Poincaré's writings in English translation庞加莱的英语翻译作品===

Popular writings on the [[philosophy of science]]:
关于[[科学哲学]]的通俗著作：

*{{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.
*1904年“科学与假设”，沃尔特·斯科特出版公司。

* 1913. "The New Mechanics," The Monist, Vol. XXIII.
*1913年。”《新力学》，《一元论》，第二十三卷。
*1913. "The Relativity of Space," The Monist, Vol. XXIII.
*1913年。”空间的相对性，《一元论》，第二十三卷。
*1913. {{Citation | title=Last Essays. |place=New York |publisher=Dover reprint, 1963 | url=https://archive.org/details/mathematicsandsc001861mbp}}
*1913年。{{引文|标题=最后一篇论文。|place=New York | publisher=Dover再版，1963 |网址=https://archive.org/details/mathematicsandsc001861mbp}}
*1956. ''Chance.'' In James R. Newman, ed., The World of Mathematics (4 Vols).
*1956年《机会》，詹姆斯·R·纽曼主编，《数学世界》（4卷）。
*1958. ''The Value of Science,'' New York: Dover.
*1958年《科学的价值》，纽约：多佛。

On [[algebraic topology]]:
关于[[代数拓扑]]：

* 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]]:
关于[[天体力学]]：

* 1892–99. ''New Methods of Celestial Mechanics'', 3 vols. English trans., 1967. {{isbn|1-56396-117-2}}.
*1892-1899年。”天体力学新方法》，3卷。英语译本，1967年。{{isbn | 1-56396-117-2}}。
* 1905. "The Capture Hypothesis of J. J. See," The Monist, Vol. XV.
*1905年。”J.J.的捕获假说，见《一元论》，第十五卷。
* 1905–10. ''Lessons of Celestial Mechanics''.
*1905-10年。”天体力学课程”。

On the [[philosophy of mathematics]]:
关于[[数学哲学]]：
* 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:

*伊瓦尔德，威廉B.，编辑，1996年从康德到希尔伯特：数学基础中的一本原著，2卷。牛津大学出版社。包含庞加莱的以下作品：
** 1894, "On the Nature of Mathematical Reasoning," 972–81.
**1894年，“论数学推理的本质”，972-81。
** 1898, "On the Foundations of Geometry," 982–1011.
**1898年，《几何基础》，982-1011年。
** 1900, "Intuition and Logic in Mathematics," 1012–20.
**1900年，“数学中的直觉和逻辑”，1012-20。
** 1905–06, "Mathematics and Logic, I–III," 1021–70.
**1905年至1906年，“数学与逻辑，I–III”，1021–70。
** 1910, "On Transfinite Numbers," 1071–74.
**1910年，“关于超限数”，1071-74。
*1905. "The Principles of Mathematical Physics," The Monist, Vol. XV.
*1905年。”《数学物理原理》，《一元论》，第十五卷。
* 1910. "The Future of Mathematics," The Monist, Vol. XX.
*1910年。”《数学的未来》，《一元论》，第二十卷。
*1910. "Mathematical Creation," The Monist, Vol. XX.
*1910年。”《数学创造》，《一元论》，第二十卷。

Other:
其他：
* 1904. ''Maxwell's Theory and Wireless Telegraphy,'' New York, McGraw Publishing Company.
*1904年《麦克斯韦理论与无线电报》，纽约，麦格劳出版公司。
* 1905. "The New Logics," The Monist, Vol. XV.
*1905年。”《新逻辑学》，《一元论》，第十五卷。
* 1905. "The Latest Efforts of the Logisticians," The Monist, Vol. XV.
*1905年。”《后勤人员的最新努力》，《一元论》，第十五卷。

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请参阅==

===Concepts概念===

* [[Poincaré complex]] – an abstraction of the singular chain complex of a closed, orientable manifold
*[[Poincarécomplex]]——封闭可定向流形的奇异链复形的抽象
* [[Poincaré duality]]
*[[庞加莱双重性]]
* [[Poincaré disk model]]
*[[庞加莱磁盘模型]]
* [[Poincaré group]]
*[[庞加莱集团]]

}}

}}

* [[Poincaré half-plane model]]
*[[庞加莱半平面模型]]
* [[Homology sphere#Poincaré homology sphere|Poincaré homology sphere]]
*[[同调球#庞加莱同调球|庞加莱同调球]]
*[[Poincaré inequality]]
*[[庞加莱不等式]]
*[[Poincaré map]]
*[[Poincarémap]]

* [[Poincaré residue]]

*[[庞加莱残留物]]
* [[Poincaré series (modular form)]]
*[[庞加莱系列（模块形式）]]
* [[Poincaré space]]
*[[庞加莱空间]]
* [[Poincaré metric]]
* [[Poincaré plot]]

* [[Hilbert–Poincaré series|Poincaré series]]
*[[希尔伯特-庞加莱系列|庞加莱系列]]
* [[Poincaré sphere (optics)|Poincaré sphere]]
*[Poincarésphere（光学）| Poincarésphere]]
* [[Poincaré–Lelong equation]]
*[Poincaré–Lelong方程]]
* [[Poincaré–Lindstedt method]]
*[Poincaré–Lindstedt方法]]
* [[Poincaré–Lindstedt perturbation theory]]
*庞加莱-林德斯特摄动理论
*[[Poincaré–Steklov operator]]
*[[Poincaré–Steklov运算符]]
* [[Reflecting Function]]
*[[反射函数]]

===Theorems理论===

|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.
*[[庞加莱递推定理]]：某些系统在足够长但有限的时间后，将返回到非常接近初始状态的状态。

|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.
*[[Poincaré–Bendixson定理]：关于平面、圆柱或两个球体上连续动力系统轨道的长期行为的陈述。

|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–Hopf定理]]：毛球定理的一个推广，它指出在没有源或汇的球体上没有光滑的向量场。

* [[Poincaré–Lefschetz duality theorem]]: a version of Poincaré duality in geometric topology, applying to a manifold with boundary
*[[Poincaré–Lefschetz对偶定理]]：几何拓扑中Poincaré对偶的一个版本，适用于有边界的流形

* [[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.
*[[庞加莱分离定理]]：给出了实对称矩阵B'AB的特征值的上下界，它可以看作是更大的实对称矩阵a在B列所跨的线性子空间上的正交投影。

* [[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定理]]：沿相反方向旋转两个边界的环空间的每个保面积、保方向同胚至少有两个不动点。

* [[Poincaré–Birkhoff–Witt theorem]]: an explicit description of the universal enveloping algebra of a Lie algebra.
*[[Poincaré–Birkhoff–Witt定理]]：李代数的泛包络代数的显式描述。

* [[Poincaré conjecture]] (now a theorem): Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
*[[Poincaré猜想]]（现在是一个定理）：每个单连通的闭3-流形都同胚于3-球面。

* [[Poincaré–Miranda theorem]]: a generalization of the [[intermediate value theorem]] to ''n'' dimensions.
*[[Poincaré–Miranda定理]]：将[[中值定理]]推广到“n”维。

=== Other其他 ===

{{Columns-list|colwidth=30em|

*[[French epistemology]]
*[[法国认识论]]
*[[History of special relativity]]
*[[狭义相对论史]]

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

*[[以亨利·彭加勒命名的事物列表]]
*[[Institut Henri Poincaré]], Paris

*[[Henri Poincaré学院]]，巴黎
*[[Brouwer fixed-point theorem]]
*[[Brouwer不动点定理]]

*[[Relativity priority dispute]]

*[[相对论优先权争议]]

*[[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>

*[[认知结构现实主义]]<ref>[http://plato.stanford.edu/entries/structural reality/#Rel“结构现实主义”]：詹姆斯·拉迪曼在“[[斯坦福哲学百科全书]]”中的词条</ref>

}}

==References参考==

===Footnotes脚注===

{{Reflist}}

===Sources资源===

* [[Eric Temple Bell|Bell, Eric Temple]], 1986. ''Men of Mathematics'' (reissue edition). Touchstone Books. {{isbn|0-671-62818-6}}.
*[[Eric Temple Bell | Bell，Eric Temple]]，1986年《数学人》（再版）。试金石书籍。{{isbn | 0-671-62818-6}}。

* Belliver, André, 1956. ''Henri Poincaré ou la vocation souveraine''. Paris: Gallimard.
*安德烈·贝利弗，1956年亨利·彭加勒·奥拉·苏维拉尼职业”。巴黎：加利玛。
*[[Peter L. Bernstein|Bernstein, Peter L]], 1996. "Against the Gods: A Remarkable Story of Risk". (p. 199–200). John Wiley & Sons.
*[[Peter L.Bernstein | Bernstein，Peter L]]，1996年。”反对上帝：一个关于风险的非凡故事”。（第199-200页）。约翰威利父子公司。
*[[Carl Benjamin Boyer|Boyer, B. Carl]], 1968. ''A History of Mathematics: Henri Poincaré'', John Wiley & Sons.
*[[Carl Benjamin Boyer | Boyer，B.Carl]]，1968年数学史：亨利庞加莱，约翰威利和儿子。
*[[IvorGrattan-Guinness|Grattan-Guinness, Ivor]], 2000. ''The Search for Mathematical Roots 1870–1940.'' Princeton Uni. Press.
*[Ivor Grattan吉尼斯| Grattan Guinness，Ivor]]，2000年1870-1940年数学根源的探索。按。
* {{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.
*{{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}}.网络版发表在2004年美国医学会杂志上。

* Folina, Janet, 1992. ''Poincaré and the Philosophy of Mathematics.'' Macmillan, New York.
*福琳娜，珍妮特，1992年彭加勒与数学哲学〉，麦克米伦，纽约。
* [[Jeremy Gray|Gray, Jeremy]], 1986. ''Linear differential equations and group theory from Riemann to Poincaré'', Birkhauser {{isbn|0-8176-3318-9}}
*1986年从黎曼到庞加莱的线性微分方程与群论，Birkhauser{isbn | 0-8176-3318-9}}
*Gray, Jeremy, 2013. ''Henri Poincaré: A scientific biography''. Princeton University Press {{isbn|978-0-691-15271-4}}
*格雷，杰里米，2013年彭加勒：科学传记”。普林斯顿大学出版社
*{{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 }}
*{{引文|网址=http://www.ams.org/notices/200509/comm-mawhin.pdf

|标题=亨利·彭加勒。为科学服务的一生

|作者=Jean Mawhin | journal=AMS通知

|日期=2005年10月|卷=52 |期=9 |页=1036–1044}

*Kolak, Daniel, 2001. ''Lovers of Wisdom'', 2nd ed. Wadsworth.
*科拉克，丹尼尔，2001年《智慧的情人》，第二版，沃兹华斯。

*Gargani, Julien, 2012. ''Poincaré, le hasard et l'étude des systèmes complexes'', L'Harmattan.
*加加尼，朱利安，2012年庞加莱，《哈萨德与系统综合体》，哈马丹。

* Murzi, 1998. "Henri Poincaré".
*穆尔齐，1998年。”亨利·彭加勒”。
*O'Connor, J. John, and Robertson, F. Edmund, 2002, "Jules Henri Poincaré". University of St. Andrews, Scotland.
*欧康纳，J.约翰和罗伯逊，F.埃德蒙，2002年，“朱尔斯-亨利-庞加莱”。苏格兰圣安德鲁斯大学。
*[[Ivars Peterson|Peterson, Ivars]], 1995. ''Newton's Clock: Chaos in the Solar System'' (reissue edition). W H Freeman & Co. {{isbn|0-7167-2724-2}}.
*[[Ivars Peterson | Peterson，Ivars]]，1995年牛顿钟：太阳系的混沌》（再版）。W H弗里曼公司{isbn | 0-7167-2724-2}。

*Sageret, Jules, 1911. ''Henri Poincaré''. Paris: Mercure de France.
*萨盖特，朱尔斯，1911年亨利·彭加勒”。巴黎：法国美爵酒店。
* Toulouse, E.,1910. ''Henri Poincaré''.—(Source biography in French) at University of Michigan Historic Math Collection.
*图卢兹，E.，1910年亨利·彭加勒（法国传记来源）密歇根大学历史数学收藏。
*{{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}}
*{引用图书| title=Mathematics and Its History | edition=3rd，插图| first1=John | last1=Stillwell | publisher=Springer Science&Business Media |年份=2010 | isbn=978-1-4419-6052-8 |网址=https://books.google.com/books？id=V7mxZqjs5yUC | ref=harv}}
*[[F. Verhulst|Verhulst, Ferdinand]], 2012 ''Henri Poincaré. Impatient Genius''. N.Y.: Springer.
*[[F.Verhulst | Verhulst，Ferdinand]]，2012年，“亨利·彭加勒。“不耐烦的天才”。N、 Y：斯普林格。

*''Henri Poincaré, l'œuvre scientifique, l'œuvre philosophique'', by Vito Volterra, Jacques Hadamard, Paul Langevin and Pierre Boutroux, Felix Alcan, 1914.
*保罗·阿苏瓦·阿库维·保罗·阿库维·费利瓦尼·波依瓦，1914年。
** ''Henri Poincaré, l'œuvre mathématique'', by [[Vito Volterra]].
**“亨利·彭加勒，l’uvre matique”，作者[[Vito Volterra]]。
** ''Henri Poincaré, le problème des trois corps'', by [[Jacques Hadamard]].
**“亨利·彭加勒，特洛伊军团的问题”，作者[[Jacques Hadamard]]。
** ''Henri Poincaré, le physicien'', by [[Paul Langevin]].
**“亨利·彭加勒，物理学”，作者[[Paul Langevin]]。
** ''Henri Poincaré, l'œuvre philosophique'', by [[Pierre Boutroux]].
**“亨利·彭加勒，l’uvre Phologique”，作者[[Pierre Boutroux]]。
*{{PlanetMath attribution|id=3793|title=Jules Henri Poincaré}}
*{{PlanetMath属性| id=3793 | title=Jules-Henri-Poincaré}}

==Further reading延伸阅读==

===Secondary sources to work on relativity是研究相对论的第二资源===

* {{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非主流来源===

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外部链接==

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 – 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

{{s-start}}

Category:Thermodynamicists

类别: 热力学家

{{s-culture}}

Category:Fluid dynamicists

类别: 流体动力学家

{{s-bef|before=[[Sully Prudhomme]]}}

Category:Topologists

类别: 拓扑学家

{{s-ttl|title=[[List of members of the Académie française#Seat 24|Seat 24]] [[Académie française]] 1908–1912|years=}}

Category:University of Paris faculty

类别: 巴黎大学教员

{{s-aft|after=[[Alfred Capus]]}}

Category:French male writers

类别: 法国男性作家

{{s-end}}

Category:Deaths from embolism

类别: 死于栓塞

{{philosophy of science}}

Category:Dynamical systems theorists

范畴: 动力系统理论家

<noinclude>

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

[[Category:待整理页面]]

2020-10-23T22:25:33Z

INCH RONG：

此词条暂由Yuling翻译，未经人工整理和审校，带来阅读不便，请见谅。

{{short description|Mathematical equation involving derivatives of an unknown function}}

{{Distinguish|Difference equation}}

[[File:Elmer-pump-heatequation.png|thumb|350px|Visualization of heat transfer in a pump casing, created by solving the [[heat equation]]. [[Heat]] is being generated internally in the casing and being cooled at the boundary, providing a [[steady state]] temperature distribution.]]

Visualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution.

通过求解热方程，建立了泵壳内传热的可视化模型。热量在外壳内部产生并在边界处冷却，从而提供稳定的温度分布。

In mathematics, a '''differential equation''' is an [[equation]] that relates one or more [[function (mathematics)|function]]s and their [[derivative]]s.<ref name="Zill2012">{{cite book|author=Dennis G. Zill|title=A First Course in Differential Equations with Modeling Applications|url=https://books.google.com/books?id=pasKAAAAQBAJ&printsec=frontcover#v=snippet&q=%22ordinary%20differential%22&f=false|date=15 March 2012|publisher=Cengage Learning|isbn=1-285-40110-7}}</ref> In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common, therefore differential equations play a prominent role in many disciplines including [[engineering]], [[physics]], [[economics]], and [[biology]].

In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common, therefore differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

在数学中，'''微分方程 Differential Equation'''是可以将一个或多个函数及其导数相互关联的方程。在实际应用中，函数通常代表物理量，导数代表其变化率，而微分方程则定义了两者之间的关系。由于这种关系十分普遍，因此微分方程在包括工程学、物理学、经济学和生物学在内的许多学科中有着突出的作用。

Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.

Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.

微分方程的研究主要包括对微分方程解(满足每个方程的函数集)及其解的性质的研究。只有最简单的微分方程才能直接用公式求解；然而，有时无需精确计算便可以确定给定微分方程的解的许多性质。

Often when a [[closed-form expression]] for the solutions is not available, solutions may be approximated numerically using computers. The theory of [[dynamical systems]] puts emphasis on [[:wikt:qualitative|qualitative]] analysis of systems described by differential equations, while many [[numerical methods]] have been developed to determine solutions with a given degree of accuracy.

Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

通常情况下，当解的封闭形式不存在时，可以用计算机进行近似计算方程的解。动力系统理论着重于对由微分方程描述的系统进行定性分析，与此同时也已经发展了许多数值方法来计算给定精度下微分方程的解。

==History==
历史

Differential equations first came into existence with the [[History of calculus|invention of calculus]] by [[Isaac Newton|Newton]] and [[Leibniz]]. In Chapter 2 of his 1671 work [[Method of Fluxions|''Methodus fluxionum et Serierum Infinitarum'']],<ref>Newton, Isaac. (c.1671). Methodus Fluxionum et Serierum Infinitarum (The Method of Fluxions and Infinite Series), published in 1736 [Opuscula, 1744, Vol. I. p. 66].</ref> Isaac Newton listed three kinds of differential equations:

Differential equations first came into existence with the invention of calculus by Newton and Leibniz. In Chapter 2 of his 1671 work Methodus fluxionum et Serierum Infinitarum, Isaac Newton listed three kinds of differential equations:

微分方程是在牛顿和莱布尼茨发明微积分后首次出现的。艾萨克 · 牛顿在他1671年的著作《无限的循环与系列》的第二章中列出了三种微分方程:

:<math>
\begin{align}
& \frac {dy}{dx} = f(x) \\[5pt]
& \frac {dy}{dx} = f(x,y) \\[5pt]
& x_1 \frac {\partial y}{\partial x_1} + x_2 \frac {\partial y}{\partial x_2} = y
\end{align}
</math>

In all these cases, {{mvar|y}} is an unknown function of {{mvar|x}} (or of <math>x_1</math> and <math>x_2</math>), and {{mvar|f}} is a given function.

In all these cases, {{mvar|y}} y is an unknown function of (or of <math>x_1</math> and <math>x_2</math>), and is a given function.

在这些情况中，{{mvar|y}}是自变量 {{mvar|x}}(或者是<math>x_1</math> 和 <math>x_2</math>)的未知函数，并且 {{mvar|f}} 是一个给定的函数。

He solves these examples and others using infinite series and discusses the non-uniqueness of solutions.

He solves these examples and others using infinite series and discusses the non-uniqueness of solutions.

他利用无穷级数求解这些例子和其他例子，并讨论了解的非唯一性。

[[Jacob Bernoulli]] proposed the [[Bernoulli differential equation]] in 1695.<ref>{{Citation | last1=Bernoulli | first1=Jacob | author1-link=Jacob Bernoulli | title=Explicationes, Annotationes & Additiones ad ea, quae in Actis sup. de Curva Elastica, Isochrona Paracentrica, & Velaria, hinc inde memorata, & paratim controversa legundur; ubi de Linea mediarum directionum, alliisque novis | year=1695 | journal=[[Acta Eruditorum]]}}</ref> This is an [[ordinary differential equation]] of the form

Jacob Bernoulli proposed the Bernoulli differential equation in 1695. This is an ordinary differential equation of the form

雅可比·伯努利在1695年提出了伯努利微分方程。这种方程是'''常微分方程 Ordinary Differential Equation'''的一种形式，

: <math>y'+ P(x)y = Q(x)y^n\,</math>

for which the following year Leibniz obtained solutions by simplifying it.<ref>{{Citation | last1=Hairer | first1=Ernst | last2=Nørsett | first2=Syvert Paul | last3=Wanner | first3=Gerhard | title=Solving ordinary differential equations I: Nonstiff problems | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-56670-0 | year=1993}}</ref>

for which the following year Leibniz obtained solutions by simplifying it.

莱布尼茨并于第二年将方程简化从而得到了方程的解。

Historically, the problem of a vibrating string such as that of a [[musical instrument]] was studied by [[Jean le Rond d'Alembert]], [[Leonhard Euler]], [[Daniel Bernoulli]], and [[Joseph-Louis Lagrange]].<ref>{{cite journal|url = http://homes.chass.utoronto.ca/~cfraser/vibration.pdf |title = Review of ''The evolution of dynamics, vibration theory from 1687 to 1742'', by John T. Cannon and Sigalia Dostrovsky|last= Frasier|first=Craig|journal=Bulletin (New Series) of the American Mathematical Society |date=July 1983 |volume= 9| issue = 1}}</ref><ref>{{cite journal |first=Gerard F. |last=Wheeler |first2=William P. |last2=Crummett |title=The Vibrating String Controversy |journal= [[American Journal of Physics|Am. J. Phys.]] |year=1987 |volume=55 |issue=1 |pages=33–37 |doi=10.1119/1.15311 |bibcode = 1987AmJPh..55...33W }}</ref><ref>For a special collection of the 9 groundbreaking papers by the three authors, see [http://www.lynge.com/item.php?bookid=38975&s_currency=EUR&c_sourcepage= First Appearance of the wave equation: D'Alembert, Leonhard Euler, Daniel Bernoulli. - the controversy about vibrating strings] (retrieved 13 Nov 2012). Herman HJ Lynge and Son.</ref><ref>For de Lagrange's contributions to the acoustic wave equation, can consult [https://books.google.com/books?id=D8GqhULfKfAC&pg=PA18 Acoustics: An Introduction to Its Physical Principles and Applications] Allan D. Pierce, Acoustical Soc of America, 1989; page 18.(retrieved 9 Dec 2012)</ref> In 1746, d’Alembert discovered the one-dimensional [[wave equation]], and within ten years Euler discovered the three-dimensional wave equation.<ref name=Speiser>Speiser, David. ''[https://books.google.com/books?id=9uf97reZZCUC&pg=PA191 Discovering the Principles of Mechanics 1600-1800]'', p. 191 (Basel: Birkhäuser, 2008).</ref>

Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange. In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation.

历史上，弦振动问题——比如乐器的弦——是由让·勒朗·达朗贝尔，欧拉，丹尼尔·伯努利和约瑟夫·路易斯·拉格朗日研究的。1746年，达朗贝尔发现了一维波动方程，10年之内，欧拉发现了三维波动方程。

The [[Euler–Lagrange equation]] was developed in the 1750s by Euler and Lagrange in connection with their studies of the [[tautochrone]] problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point. Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to [[mechanics]], which led to the formulation of [[Lagrangian mechanics]].

The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point. Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to mechanics, which led to the formulation of Lagrangian mechanics.

欧拉-拉格朗日方程式是欧拉和拉格朗日在18世纪50年代结合他们对等时降线问题的研究而发明的。这是一个不考虑起始点的求解曲线问题，问题中一个加权的粒子将在一个固定的时间内下降到一个固定的点。拉格朗日在1755年解决了这个问题，并将其发送给欧拉。两者都进一步发展了拉格朗日的方法并将其应用于力学，从而促使了拉格朗日力学的形成。
==[[用户:Yuling|Yuling]]（[[用户讨论:Yuling|讨论]]） independent of the starting point 这里翻译不太好。

In 1822, [[Joseph Fourier|Fourier]] published his work on [[heat flow]] in ''Théorie analytique de la chaleur'' (The Analytic Theory of Heat),<ref>{{Cite book | last = Fourier | first = Joseph | title = Théorie analytique de la chaleur | publisher = Firmin Didot Père et Fils | year = 1822 | location = Paris | language = French | url=https://archive.org/details/bub_gb_TDQJAAAAIAAJ | oclc=2688081 }}</ref> in which he based his reasoning on [[Newton's law of cooling]], namely, that the flow of heat between two adjacent molecules is proportional to the extremely small difference of their temperatures. Contained in this book was Fourier's proposal of his [[heat equation]] for conductive diffusion of heat. This partial differential equation is now taught to every student of mathematical physics.

In 1822, Fourier published his work on heat flow in Théorie analytique de la chaleur (The Analytic Theory of Heat), in which he based his reasoning on Newton's law of cooling, namely, that the flow of heat between two adjacent molecules is proportional to the extremely small difference of their temperatures. Contained in this book was Fourier's proposal of his heat equation for conductive diffusion of heat. This partial differential equation is now taught to every student of mathematical physics.

1822年，傅立叶在《热的分析理论》中发表了他关于热流的研究工作，其中他以牛顿的冷却定律为基础进行推理，即两个相邻分子之间的热流与它们之间极小的温差成正比。这本书中包含了傅立叶关于热传导扩散的热方程式的建议。现在，每一个学习数学物理的学生都需要学习这类偏微分方程。

==Example==
示例

In [[classical mechanics]], the motion of a body is described by its position and velocity as the time value varies. [[Newton's laws of motion|Newton's laws]] allow these variables to be expressed dynamically (given the position, velocity, acceleration and various forces acting on the body) as a differential equation for the unknown position of the body as a function of time.

In classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow these variables to be expressed dynamically (given the position, velocity, acceleration and various forces acting on the body) as a differential equation for the unknown position of the body as a function of time.

在经典力学中，物体的运动是由其不断随时间变化的位置和速度来描述的。这些变量的表达在牛顿定律中是动态的(给定位置、速度、加速度和作用在物体上的各种力) ，并以时间函数的形式给出了未知物体位置的微分方程。

In some cases, this differential equation (called an [[equations of motion|equation of motion]]) may be solved explicitly.

In some cases, this differential equation (called an equation of motion) may be solved explicitly.

在某些情况下，这种微分方程(称为运动方程)可以精确地求解。

An example of modeling a real-world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This means that the ball's acceleration, which is a derivative of its velocity, depends on the velocity (and the velocity depends on time). Finding the velocity as a function of time involves solving a differential equation and verifying its validity.

An example of modeling a real-world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This means that the ball's acceleration, which is a derivative of its velocity, depends on the velocity (and the velocity depends on time). Finding the velocity as a function of time involves solving a differential equation and verifying its validity.

使用微分方程模拟现实世界问题的一个例子是仅考虑重力和空气阻力来确定球在空中落下的速度。球对地面的加速度是由于重力加速度减去由于空气阻力提供的加速度。重力被认为是常数，空气阻力可以被模拟为与球的速度成正比。这意味着球的加速度，也就是其速度的导数，取决于速度(而速度取决于时间)。找到时间的函数--速度--需要解决一个微分方程问题并验证其正确性。

==Types==
微分方程的类型

Differential equations can be divided into several types. Apart from describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. Commonly used distinctions include whether the equation is: Ordinary/Partial, Linear/Non-linear, and Homogeneous/heterogeneous. This list is far from exhaustive; there are many other properties and subclasses of differential equations which can be very useful in specific contexts.

Differential equations can be divided into several types. Apart from describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. Commonly used distinctions include whether the equation is: Ordinary/Partial, Linear/Non-linear, and Homogeneous/heterogeneous. This list is far from exhaustive; there are many other properties and subclasses of differential equations which can be very useful in specific contexts.

微分方程可分为几种类型。除了描述方程本身的性质之外，微分方程的类型有助于指导选择何种解决方案。常见的区别包括方程是否为: 常微分/偏微分方程、线性/非线性方程和齐次/非齐次方程。这份清单还很长，微分方程还有许多在特定的情况下非常有用的其他性质和子类。

===Ordinary differential equations===
常微分方程
{{main|Ordinary differential equation|Linear differential equation}}

An [[ordinary differential equation]] (''ODE'') is an equation containing an unknown [[function of a real variable|function of one real or complex variable]] {{mvar|x}}, its derivatives, and some given functions of {{mvar|x}}. The unknown function is generally represented by a [[variable (mathematics)|variable]] (often denoted {{mvar|y}}), which, therefore, ''depends'' on {{mvar|x}}. Thus {{mvar|x}} is often called the [[independent variable]] of the equation. The term "''ordinary''" is used in contrast with the term [[partial differential equation]], which may be with respect to ''more than'' one independent variable.

An ordinary differential equation (ODE) is an equation containing an unknown function of one real or complex variable , its derivatives, and some given functions of . The unknown function is generally represented by a variable (often denoted {{mvar|y}}), which, therefore, depends on {{mvar|x}}. Thus {{mvar|x}} is often called the independent variable of the equation. The term "ordinary" is used in contrast with the term partial differential equation, which may be with respect to more than one independent variable.

常微分方程是只含有一个实变量或复变量的未知函数，其导数以及此函数的一些方程。未知函数通常由一个变量(通常由 {{mvar|y}} 表示)表示，因此这个变量依赖于 {{mvar|x}} 。因此 {{mvar|x}} 通常被称为方程式的自变量。“常微分方程”一词与偏微分方程一词相比，后者涉及一个以上的独立变量。

[[Linear differential equation]]s are the differential equations that are [[linear equation|linear]] in the unknown function and its derivatives. Their theory is well developed, and in many cases one may express their solutions in terms of [[antiderivative|integrals]].

Linear differential equations are the differential equations that are linear in the unknown function and its derivatives. Their theory is well developed, and in many cases one may express their solutions in terms of integrals.

线性微分方程是指方程中未知函数及其导数都是线性的微分方程。关于这些方程的理论发展得很好，在多数情况下可以用积分来表示他们的解。

Most ODEs that are encountered in [[physics]] are linear. Therefore, most [[special functions]] may be defined as solutions of linear differential equations (see [[Holonomic function]]).

Most ODEs that are encountered in physics are linear. Therefore, most special functions may be defined as solutions of linear differential equations (see Holonomic function).

物理学中遇到的大多数常微分方程都是线性的。因此，大多数特殊函数可以定义为线性微分方程的解(见完整性函数)。

As, in general, the solutions of a differential equation cannot be expressed by a [[closed-form expression]], [[numerical ordinary differential equations|numerical methods]] are commonly used for solving differential equations on a computer.

As, in general, the solutions of a differential equation cannot be expressed by a closed-form expression, numerical methods are commonly used for solving differential equations on a computer.

一般来说，微分方程的解不能用解析解表示，而通常会在计算机上利用数值方法求解微分方程。

===Partial differential equations===
偏微分方程
{{main|Partial differential equation}}

A [[partial differential equation]] (''PDE'') is a differential equation that contains unknown [[Multivariable calculus|multivariable function]]s and their [[partial derivatives]]. (This is in contrast to [[ordinary differential equations]], which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved in closed form, or used to create a relevant [[computer model]].

A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. (This is in contrast to ordinary differential equations, which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved in closed form, or used to create a relevant computer model.

'''偏微分方程 Partial Differential Equation'''是一种包含多元函数及其偏导数的微分方程函数(这与处理单变量函数及其导数的常微分方程不同)。偏微分方程用于描述涉及多元函数的问题，或者以封闭形式求解，或者用于创建相关的计算机模型。

PDEs can be used to describe a wide variety of phenomena in nature such as [[sound]], [[heat]], [[electrostatics]], [[electrodynamics]], [[fluid flow]], [[Elasticity (physics)|elasticity]], or [[quantum mechanics]]. These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional [[dynamical systems]], partial differential equations often model [[multidimensional systems]]. [[Stochastic partial differential equations]] generalize partial differential equations for modeling [[randomness]].

PDEs can be used to describe a wide variety of phenomena in nature such as sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalized similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. Stochastic partial differential equations generalize partial differential equations for modeling randomness.

偏微分方程可以用来描述自然界中各种各样的现象，如声音、热量、静电、电动力学、流体流动、弹性或量子力学。这些看起来截然不同的物理现象可以用相似的偏微分方程表达。正如常微分方程经常对一维动力系统进行建模一样，偏微分方程经常对多维系统进行建模。随机偏微分方程延伸了偏微分方程在模拟随机性上的应用。

===Non-linear differential equations===
非线性微分方程

{{main|Non-linear differential equations}}

A '''non-linear differential equation''' is a differential equation that is not a [[linear equation]] in the unknown function and its derivatives (the linearity or non-linearity in the arguments of the function are not considered here). There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular [[Symmetry|symmetries]]. Nonlinear differential equations can exhibit very complicated behavior over extended time intervals, characteristic of [[chaos theory|chaos]]. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. [[Navier–Stokes existence and smoothness]]). However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.

A non-linear differential equation is a differential equation that is not a linear equation in the unknown function and its derivatives (the linearity or non-linearity in the arguments of the function are not considered here). There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular symmetries. Nonlinear differential equations can exhibit very complicated behavior over extended time intervals, characteristic of chaos. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. Navier–Stokes existence and smoothness). However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.

非线性微分方程是微分方程的一种，但它不是关于未知函数及其导数的线性方程(这里不考虑函数本身的线性或非线性)。能够精确求解非线性微分方程的方法很少; 那些已有的方法通常依赖于方程具有某种特定的对称性。非线性微分方程在更长的时间段内表现出非常复杂的行为，具有混沌特性。即使是非线性微分方程解的存在性、唯一性和可扩展性等基本问题，以及非线性偏微分方程初边值问题的适定性问题，也是一个难题（查阅，纳维-斯托克斯方程的存在性和光滑性)。然而，如果微分方程是一个有意义物理过程的正确表述，那么人们期望它有一个解析解。

Linear differential equations frequently appear as [[linearization|approximations]] to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations (see below).

Linear differential equations frequently appear as approximations to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations (see below).

线性微分方程经常作为非线性方程的近似形式出现。这些近似只有某些限制条件下才有效。例如，谐振子方程是非线性摆方程的近似这一情况只有对于小幅度振荡是有效的(见下文)。

==={{anchor|Second order}} Equation order===
方程的阶

Differential equations are described by their order, determined by the term with the [[Derivative#Higher derivatives|highest derivatives]]. An equation containing only first derivatives is a ''first-order differential equation'', an equation containing the [[second derivative]] is a ''second-order differential equation'', and so on.<ref>[[Eric W Weisstein|Weisstein, Eric W]]. "Ordinary Differential Equation Order." From [[MathWorld]]--A Wolfram Web Resource. http://mathworld.wolfram.com/OrdinaryDifferentialEquationOrder.html</ref><ref>[http://www.kshitij-iitjee.com/Maths/Differential-Equations/order-and-degree-of-a-differential-equation.aspx Order and degree of a differential equation] {{Webarchive|url=https://web.archive.org/web/20160401070512/http://www.kshitij-iitjee.com/Maths/Differential-Equations/order-and-degree-of-a-differential-equation.aspx |date=2016-04-01 }}, accessed Dec 2015.</ref> Differential equations that describe natural phenomena almost always have only first and second order derivatives in them, but there are some exceptions, such as the [[Thin-film equation|thin film equation]], which is a fourth order partial differential equation.

Differential equations are described by their order, determined by the term with the highest derivatives. An equation containing only first derivatives is a first-order differential equation, an equation containing the second derivative is a second-order differential equation, and so on. Differential equations that describe natural phenomena almost always have only first and second order derivatives in them, but there are some exceptions, such as the thin film equation, which is a fourth order partial differential equation.

微分方程是用它们的阶来描述的，并且是由导数最高的项来确定。只含有一阶导数的方程是一阶微分方程，含有二阶导数的方程是二阶微分方程，等等。描述自然现象的微分方程几乎总是只有一阶和二阶导数，但也有一些例外，例如薄膜方程，它是一个四阶偏微分方程。

===Examples===
示例

In the first group of examples ''u'' is an unknown function of ''x'', and ''c'' and ''ω'' are constants that are supposed to be known. Two broad classifications of both ordinary and partial differential equations consist of distinguishing between ''[[linear differential equation|linear]]'' and ''nonlinear'' differential equations, and between [[homogeneous differential equation|''homogeneous'' differential equation]]s and ''heterogeneous'' ones.

In the first group of examples u is an unknown function of x, and c and ω are constants that are supposed to be known. Two broad classifications of both ordinary and partial differential equations consist of distinguishing between linear and nonlinear differential equations, and between homogeneous differential equations and heterogeneous ones.

在第一组例子中，待求解的''u''是''x''的函数，''c''和''ω''是应该已知的常数。常微分方程和偏微分方程这两种广义分类下还要区分微分方程的线性和非线性，以及区分微分方程的齐次和非齐次。

* Heterogeneous first-order linear constant coefficient ordinary differential equation:
非齐次一阶常系数常微分方程

:: <math> \frac{du}{dx} = cu+x^2. </math>

* Homogeneous second-order linear ordinary differential equation:
齐次二阶线性常微分方程

:: <math> \frac{d^2u}{dx^2} - x\frac{du}{dx} + u = 0. </math>

* Homogeneous second-order linear constant coefficient ordinary differential equation describing the [[harmonic oscillator]]:
用于描述简谐振动的齐次二阶常系数常系数微分方程

:: <math> \frac{d^2u}{dx^2} + \omega^2u = 0. </math>

* Heterogeneous first-order nonlinear ordinary differential equation:
非齐次一阶非线性常微分方程

:: <math> \frac{du}{dx} = u^2 + 4. </math>

* Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a [[pendulum]] of length ''L'':
用于描述摆长为L的钟摆运动的二阶非线性（因正弦函数产生）常微分方程

:: <math> L\frac{d^2u}{dx^2} + g\sin u = 0. </math>

In the next group of examples, the unknown function ''u'' depends on two variables ''x'' and ''t'' or ''x'' and ''y''.

In the next group of examples, the unknown function u depends on two variables x and t or x and y.

在下一组例子中，未知函数''u''依赖于两个变量''x'' 和 ''t''或者''x''和''y''。

* Homogeneous first-order linear partial differential equation:
齐次一阶线性偏微分方程

:: <math> \frac{\partial u}{\partial t} + t\frac{\partial u}{\partial x} = 0. </math>

* Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the [[Laplace equation]]:
齐次二阶线性常系数椭圆形偏微分方程，也称为拉普拉斯方程

:: <math> \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0. </math>

* Homogeneous third-order non-linear partial differential equation :
齐次三阶非线性偏微分方程

:: <math> \frac{\partial u}{\partial t} = 6u\frac{\partial u}{\partial x} - \frac{\partial^3 u}{\partial x^3}. </math>

==Existence of solutions==
解的存在性

Solving differential equations is not like solving [[algebraic equations]]. Not only are their solutions often unclear, but whether solutions are unique or exist at all are also notable subjects of interest.

Solving differential equations is not like solving algebraic equations. Not only are their solutions often unclear, but whether solutions are unique or exist at all are also notable subjects of interest.

解微分方程不同于解代数方程。不仅它们的解往往不清楚，而且解是否唯一或是否存在也是值得关注的课题。

For first order initial value problems, the [[Peano existence theorem]] gives one set of circumstances in which a solution exists. Given any point <math>(a,b)</math> in the xy-plane, define some rectangular region <math>Z</math>, such that <math>Z = [l,m]\times[n,p]</math> and <math>(a,b)</math> is in the interior of <math>Z</math>. If we are given a differential equation <math>\frac{dy}{dx} = g(x,y)</math> and the condition that <math>y=b</math> when <math>x=a</math>, then there is locally a solution to this problem if <math>g(x,y)</math> and <math>\frac{\partial g}{\partial x}</math> are both continuous on <math>Z</math>. This solution exists on some interval with its center at <math>a</math>. The solution may not be unique. (See [[Ordinary differential equation]] for other results.)

For first order initial value problems, the Peano existence theorem gives one set of circumstances in which a solution exists. Given any point <math>(a,b)</math> in the xy-plane, define some rectangular region <math>Z</math>, such that <math>Z = [l,m]\times[n,p]</math> and <math>(a,b)</math> is in the interior of <math>Z</math>. If we are given a differential equation <math>\frac{dy}{dx} = g(x,y)</math> and the condition that <math>y=b</math> when <math>x=a</math>, then there is locally a solution to this problem if <math>g(x,y)</math> and <math>\frac{\partial g}{\partial x}</math> are both continuous on <math>Z</math>. This solution exists on some interval with its center at <math>a</math>. The solution may not be unique. (See Ordinary differential equation for other results.)

对于一阶初值问题，皮亚诺存在性定理给出了一组解存在的情况。给定的x-y平面上的任意点 <math>(a,b)</math> ，定义一些矩形区域 <math>Z</math> ，比如说，<math>Z = [l,m]\times[n,p]</math> 而且 <math>(a,b)</math> 是 <math>Z</math> 内部一点。如果我们给出一个微分方程 <math>\frac{dy}{dx} = g(x,y)</math> 和当<math>x=a</math>时<math>y=b</math>，如果<math>g(x,y)</math>和<math>\frac{\partial g}{\partial x}</math>在<math>Z</math>上是连续的，这个问题就有一个局部解。这个解在以 <math>a</math> 为中心的某些区间上存在，其可能不是唯一的。(其他结果请参见常微分方程。)

However, this only helps us with first order [[initial value problem]]s. Suppose we had a linear initial value problem of the nth order:

However, this only helps us with first order initial value problems. Suppose we had a linear initial value problem of the nth order:

然而，这只能帮助我们解决一阶初始值问题。假设我们有一个n阶线性初始值问题:

:<math>
f_{n}(x)\frac{d^n y}{dx^n} + \cdots + f_{1}(x)\frac{d y}{dx} + f_{0}(x)y = g(x)
</math>

such that

such that

其中有

:<math>
y(x_{0})=y_{0}, y'(x_{0}) = y'_{0}, y''(x_{0}) = y''_{0}, \cdots
</math>

For any nonzero <math>f_{n}(x)</math>, if <math>\{f_{0},f_{1},\cdots\}</math> and <math>g</math> are continuous on some interval containing <math>x_{0}</math>, <math>y</math> is unique and exists.<ref>{{cite book|last1=Zill|first1=Dennis G.|title=A First Course in Differential Equations|publisher=Brooks/Cole|isbn=0-534-37388-7|edition=5th|year=2001}}</ref>

For any nonzero <math>f_{n}(x)</math>, if <math>\{f_{0},f_{1},\cdots\}</math> and <math>g</math> are continuous on some interval containing <math>x_{0}</math>, <math>y</math> is unique and exists.

对于任意非零的 <math>f_{n}(x)</math> ，如果<math>\{f_{0},f_{1},\cdots\}</math> 和 <math>g</math>在某个包含<math>x_{0}</math>的区间上连续，则<math>y</math>是存在且唯一的。

==Related concepts==
相关概念

* A [[delay differential equation]] (DDE) is an equation for a function of a single variable, usually called '''time''', in which the derivative of the function at a certain time is given in terms of the values of the function at earlier times.
延迟微分方程（DDE）是一元函数的方程，变量通常为时间，其中函数在一定时间点的微分会被较早时间点的函数值表达。

*An [[integro-differential equation]] (IDE) is an equation that combines aspects of a differential equation and an [[integral equation]].
积分微分方程（IDE）结合了微分方程和积分方程。

* A [[stochastic differential equation]] (SDE) is an equation in which the unknown quantity is a [[stochastic process]] and the equation involves some known stochastic processes, for example, the [[Wiener process]] in the case of diffusion equations.
随机微分方程（SDE）中的未知量是处于随机过程，并且该方程涉及一些已知的随机过程，例如，扩散方程中的维纳过程。

*A [[stochastic partial differential equation]] (SPDE) is an equation that generalizes SDEs to include space-time noise processes, with applications in [[quantum field theory]] and [[statistical mechanics]].
随机偏微分方程（SPDE）是一种包含有空间时间噪声过程的广义随机微分方程，它通常应用于量子场论以及统计力学中。

* A [[differential algebraic equation]] (DAE) is a differential equation comprising differential and algebraic terms, given in implicit form.
微分代数方程（DAE）是一种包含微分和代数项的微分方程，以隐式形式给出。

==Connection to difference equations==
与差分方程之间的联系
{{See also|Time scale calculus}}

The theory of differential equations is closely related to the theory of [[difference equations]], in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve the approximation of the solution of a differential equation by the solution of a corresponding difference equation.

The theory of differential equations is closely related to the theory of difference equations, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve the approximation of the solution of a differential equation by the solution of a corresponding difference equation.

微分方程理论与差分方程理论密切相关，在差分方程理论中，坐标系中只假定存在离散值，计算中会涉及到未知函数或已知函数的值以及坐标附近的值。许多计算微分方程数值解或研究微分方程性质的方法都会涉及到通过相应差分方程的解来逼近微分方程的解这一过程。

==Applications==

The study of differential equations is a wide field in [[pure mathematics|pure]] and [[applied mathematics]], [[physics]], and [[engineering]]. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modeling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have [[closed-form expression|closed form]] solutions. Instead, solutions can be approximated using [[Numerical ordinary differential equations|numerical methods]].

The study of differential equations is a wide field in pure and applied mathematics, physics, and engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modeling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods.

微分方程的研究涉及很多领域，例如理论数学，应用数学，物理和工程。所有这些学科都与各种类型的微分方程的性质有关。纯数学关注解的存在性和唯一性，而应用数学则强调解的逼近方法的严格性。从天体运动到桥梁设计，再到神经元之间的相互作用，微分方程在几乎所有物理、技术或生物过程的建模中都扮演着重要的角色。那些用来解决实际问题的微分方程，可能不一定是直接可解的，比如它们没有封闭形式的解。但取而代之的是，我们可以用数值方法来近似方程的解。

Many fundamental laws of [[physics]] and [[chemistry]] can be formulated as differential equations. In [[biology]] and [[economics]], differential equations are used to [[mathematical modelling|model]] the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider the propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order [[partial differential equation]], the [[wave equation]], which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by [[Joseph Fourier]], is governed by another second-order partial differential equation, the [[heat equation]]. It turns out that many [[diffusion]] processes, while seemingly different, are described by the same equation; the [[Black–Scholes]] equation in finance is, for instance, related to the heat equation.

Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider the propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order partial differential equation, the wave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by Joseph Fourier, is governed by another second-order partial differential equation, the heat equation. It turns out that many diffusion processes, while seemingly different, are described by the same equation; the Black–Scholes equation in finance is, for instance, related to the heat equation.

许多物理和化学的基本定律可以表述为微分方程。在生物学和经济学中，微分方程被用来模拟复杂系统的行为。微分方程的数学理论最初是与微分方程起源于和得到应用的科学一起发展起来。然而，有时完全不同的科学领域，却可能产生相同的微分方程。每当这种情况发生时，方程后面的数学理论可以被看作是不同现象背后的统一原则。例如，考虑光和声在大气中的传播，以及池塘表面的波的传播。所有这些都可以用相同的二阶偏微分方程来描述，即波动方程，它允许我们把光和声音想象成波的形式，很像水中相似的波。热传导的理论是由约瑟夫.傅里叶提出的，这一过程由另一个二阶偏微分方程——热方程所支配。事实证明，许多扩散过程，虽然看起来不同，却用同一个方程来描述; 例如，金融学中的布莱克-斯科尔斯方程就与热方程有关。

==[[用户:Yuling|Yuling]]（[[用户讨论:Yuling|讨论]]） "results found application" 翻译为“方程解的搜索”，可能不太准确

The number of differential equations that have received a name, in various scientific areas is a witness of the importance of the topic. See [[List of named differential equations]].

The number of differential equations that have received a name, in various scientific areas is a witness of the importance of the topic. See List of named differential equations.

事实上，同一类型的微分方程可以应用于不同领域这样的现象屡见不鲜，这足以证明微分方程这一课题的重要性。参见已命名的微分方程列表。

==See also==

{{Div col|colwidth=22em}}

*[[Complex differential equation]]
复微分方程
*[[Exact differential equation]]
精确微分方程
*[[Functional differential equation]]
泛函微分方程
*[[Initial condition]]
初始条件
*[[Integral equations]]
积分方程
*[[Numerical methods for ordinary differential equations]]
求解常微分方程的数值方法
*[[Numerical methods for partial differential equations]]
求解偏微分方程的数值方法
*[[Picard–Lindelöf theorem]] on existence and uniqueness of solutions
关于解的存在性和唯一性的皮卡德–林德洛夫定理
*[[Recurrence relation]], also known as 'difference equation'
递推关系，也称为差分方程
*[[Abstract differential equation]]
抽象微分方程
*[[System of differential equations]]
微分方程组
{{div col end}}

==References==

{{reflist|30em}}

==Further reading==

*{{cite book |first=P. |last=Abbott |first2=H. |last2=Neill |title=Teach Yourself Calculus |year=2003 |pages=266–277 }}

*{{cite book |first=P. |last=Blanchard |authorlink2=Robert L. Devaney |first2=R. L. |last2=Devaney |first3=G. R. |last3=Hall |title=Differential Equations |location= |publisher=Thompson |year=2006 }}

*{{cite book |first=W. |last=Boyce | first2=R. |last2=DiPrima| first3=D. |last3=Meade| title=Elementary Differential Equations and Boundary Value Problems|publisher=Wiley |year=2017}}

*{{cite book |first=E. A. |last=Coddington |first2=N. |last2=Levinson |title=Theory of Ordinary Differential Equations |url=https://archive.org/details/theoryofordinary00codd |url-access=registration |publisher=McGraw-Hill |year=1955 }}

*{{cite book |first=E. L. |last=Ince |title=Ordinary Differential Equations |publisher=Dover |year=1956 }}

*{{cite book |first=W. |last=Johnson |url=http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=abv5010.0001.001 |title=A Treatise on Ordinary and Partial Differential Equations |publisher=John Wiley and Sons |year=1913 }} In [http://hti.umich.edu/u/umhistmath/ University of Michigan Historical Math Collection]

*{{cite book |first=A. D. |last=Polyanin |first2=V. F. |last2=Zaitsev |title=Handbook of Exact Solutions for Ordinary Differential Equations |edition=2nd |publisher=Chapman & Hall/CRC Press |location=Boca Raton |year=2003 |isbn=1-58488-297-2 }}

*{{cite book |first=R. I. |last=Porter |title=Further Elementary Analysis |year=1978 |chapter=XIX Differential Equations }}

*{{Cite book| last = Teschl| given = Gerald|authorlink=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]]| year = 2012| isbn= 978-0-8218-8328-0| url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}

*{{cite book|author=Daniel Zwillinger|title=Handbook of Differential Equations|url=https://books.google.com/?id=n7TiBQAAQBAJ&printsec=frontcover&dq=%22Handbook+of+Differential+Equations%22#v=onepage&q=%22Handbook%20of%20Differential%20Equations%22&f=false|date=12 May 2014|publisher=Elsevier Science|isbn=978-1-4832-6396-0}}

==External links==

{{wikiquote}}

{{Wikibooks|Ordinary Differential Equations}}

{{Wikiversity|Differential equations}}

{{EB1911 poster|Differential Equation}}

*{{Commonscatinline|Differential equations}}

*[http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/ Lectures on Differential Equations] [[MIT]] Open CourseWare Videos

*[http://tutorial.math.lamar.edu/classes/de/de.aspx Online Notes / Differential Equations] Paul Dawkins, [[Lamar University]]

*[http://www.sosmath.com/diffeq/diffeq.html Differential Equations], S.O.S. Mathematics

*[http://www.fioravante.patrone.name/mat/u-u/en/differential_equations_intro.htm Introduction to modeling via differential equations] Introduction to modeling by means of differential equations, with critical remarks.

*[http://user.mendelu.cz/marik/maw/index.php?lang=en&form=ode Mathematical Assistant on Web] Symbolic ODE tool, using [[Maxima (software)|Maxima]]

*[http://eqworld.ipmnet.ru/en/solutions/ode.htm Exact Solutions of Ordinary Differential Equations]

*[http://www.hedengren.net/research/models.htm Collection of ODE and DAE models of physical systems] MATLAB models

*[http://www.jirka.org/diffyqs/ Notes on Diffy Qs: Differential Equations for Engineers] An introductory textbook on differential equations by Jiri Lebl of [[UIUC]]

*[http://www.khanacademy.org/math/differential-equations Khan Academy Video playlist on differential equations ] Topics covered in a first year course in differential equations.

*[https://web.archive.org/web/20130607120716/http://math.rareinfos.com/category/courses/solutions-differential-equations/homogeneous-linear-systems/ MathDiscuss Video playlist on differential equations ]

{{Areas of mathematics | state=collapsed}}

{{Differential equations topics}}

{{Authority control}}

[[Category:Differential equations| ]]

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This page was moved from [[wikipedia:en:Differential equation]]. Its edit history can be viewed at [[微分方程/edithistory]]</noinclude>

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