Editing Henri Poincaré (section)

==Work==

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

Among the specific topics he contributed to are the following:
*[[algebraic topology]] (a field that Poincaré virtually invented)
*the theory of analytic functions of [[several complex variables]]
*[[abelian variety|the theory of abelian functions]]
*[[algebraic geometry]]
*the [[Poincaré conjecture]], proven in 2003 by [[Grigori Perelman]].
*[[Poincaré recurrence theorem]]
*[[hyperbolic geometry]]
*[[number theory]]
*the [[three-body problem]]
*the theory of [[diophantine equation]]s
*[[electromagnetism]]
*[[special relativity]]
*the [[fundamental group]]
*In the field of [[differential equations]] Poincaré has given many results that are critical for the qualitative theory of differential equations, for example the [[Poincaré homology sphere|Poincaré sphere]] and the [[Poincaré map]].
*Poincaré on "everybody's belief" in the [[q:Henri Poincaré|''Normal Law of Errors'']] (see [[normal distribution]] for an account of that "law")
*Published an influential paper providing a novel mathematical argument in support of [[quantum mechanics]].<ref name=McCormmach>{{Citation | last = McCormmach | first = Russell | title = Henri Poincaré and the Quantum Theory | journal = Isis | volume = 58 | issue = 1 | pages = 37–55 | date =Spring 1967 | doi =10.1086/350182| s2cid = 120934561 }}</ref><ref name=Irons>{{Citation | last = Irons | first = F. E. | title = Poincaré's 1911–12 proof of quantum discontinuity interpreted as applying to atoms | journal = American Journal of Physics | volume = 69 | issue = 8 | pages = 879–884 | date = August 2001 | doi =10.1119/1.1356056 |bibcode = 2001AmJPh..69..879I }}</ref>

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

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

In case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was finally awarded to Poincaré, even though he did not solve the original problem. One of the judges, the distinguished [[Karl Weierstrass]], said, ''"This work cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that its publication will inaugurate a new era in the history of celestial mechanics."'' (The first version of his contribution even contained a serious error; for details see the article by Diacu<ref name=diacu>{{Citation
| last=Diacu|first= Florin | year=1996 | title=The solution of the ''n''-body Problem | journal=The Mathematical Intelligencer | volume =18 | pages =66–70 | doi=10.1007/BF03024313
| issue=3|s2cid= 119728316 }}</ref> and the book by [[June Barrow-Green|Barrow-Green]]<ref>{{Cite book|title=Poincaré and the three body problem|title-link= Poincaré and the Three-Body Problem |last=Barrow-Green|first=June|publisher=[[American Mathematical Society]]|year=1997|isbn=978-0821803677|location=Providence, RI|series=History of Mathematics|volume=11|oclc=34357985}}</ref>). The version finally printed<ref>{{Cite book|title=The three-body problem and the equations of dynamics: Poincaré's foundational work on dynamical systems theory|last=Poincaré|first=J. Henri|publisher=Springer International Publishing|others=Popp, Bruce D. (Translator)|year=2017|isbn=9783319528984|location=Cham, Switzerland|oclc=987302273}}</ref> contained many important ideas which led to the [[chaos theory|theory of chaos]]. The problem as stated originally was finally solved by [[Karl F. Sundman]] for ''n''&nbsp;=&nbsp;3 in 1912 and was generalised to the case of ''n''&nbsp;>&nbsp;3 bodies by [[Qiudong Wang]] in the 1990s. The series solutions have very slow convergence. It would take millions of terms to determine the motion of the particles for even very short intervals of time, so they are unusable in numerical work.<ref name="diacu" />

===Work on relativity===
{{Main|Lorentz ether theory|History of special relativity}}
[[Image:Curie and Poincare 1911 Solvay.jpg|thumb|right|[[Marie Curie]] and Poincaré talk at the 1911 [[Solvay Conference]].]]

====Local time====
Poincaré's work at the Bureau des Longitudes on establishing international time zones led him to consider how clocks at rest on the Earth, which would be moving at different speeds relative to absolute space (or the "[[luminiferous aether]]"), could be synchronised. At the same time Dutch theorist [[Hendrik Lorentz]] was developing Maxwell's theory into a theory of the motion of charged particles ("electrons" or "ions"), and their interaction with radiation. In 1895 Lorentz had introduced an auxiliary quantity (without physical interpretation) called "local time" <math>t^\prime = t-v x/c^2 \,</math><ref>{{Citation|title=A broader view of relativity: general implications of Lorentz and Poincaré invariance|volume=10|first1=Jong-Ping|last1=Hsu|first2=Leonardo|last2=Hsu|publisher=World Scientific|year=2006|isbn=978-981-256-651-5|page=37
|url=https://books.google.com/books?id=amLqckyrvUwC}}, [https://books.google.com/books?id=amLqckyrvUwC&pg=PA37 Section A5a, p 37]</ref> and introduced the hypothesis of [[length contraction]] to explain the failure of optical and electrical experiments to detect motion relative to the aether (see [[Michelson–Morley experiment]]).<ref>{{Citation
| last=Lorentz|first= Hendrik A. | author-link=Hendrik Lorentz| year=1895 | title=Versuch einer theorie der electrischen und optischen erscheinungen in bewegten Kõrpern | place =Leiden| publisher=E.J. Brill| title-link=s:de:Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern }}</ref> Poincaré was a constant interpreter (and sometimes friendly critic) of Lorentz's theory. Poincaré as a philosopher was interested in the "deeper meaning". Thus he interpreted Lorentz's theory and in so doing he came up with many insights that are now associated with special relativity. In [[s:The Measure of Time|The Measure of Time]] (1898), Poincaré said, "A little reflection is sufficient to understand that all these affirmations have by themselves no meaning. They can have one only as the result of a convention." He also argued that scientists have to set the constancy of the speed of light as a [[postulate]] to give physical theories the simplest form.<ref>{{Citation
| last=Poincaré|first= Henri | year=1898 | title=The Measure of Time | journal=Revue de Métaphysique et de Morale | volume =6 | pages =1–13| title-link=s:The Measure of Time }}</ref>
Based on these assumptions he discussed in 1900 Lorentz's "wonderful invention" of local time and remarked that it arose when moving clocks are synchronised by exchanging light signals assumed to travel with the same speed in both directions in a moving frame.<ref name=action>{{Citation
| last=Poincaré|first= Henri | year=1900 | title=La théorie de Lorentz et le principe de réaction | journal=Archives Néerlandaises des Sciences Exactes et Naturelles | volume =5 | pages =252–278| title-link=s:fr:La théorie de Lorentz et le principe de réaction }}. See also the [http://www.physicsinsights.org/poincare-1900.pdf English translation]</ref>

====Principle of relativity and Lorentz transformations====
{{Further|History of Lorentz transformations}}

In 1881 Poincaré described [[hyperbolic geometry]] in terms of the [[hyperboloid model]], formulating transformations leaving invariant the [[Lorentz interval]] <math>x^2+y^2-z^2=-1</math>, which makes them mathematically equivalent to the Lorentz transformations in 2+1 dimensions.<ref>{{Cite journal|author=Poincaré, H.|year=1881|title=Sur les applications de la géométrie non-euclidienne à la théorie des formes quadratiques|journal=Association Française Pour l'Avancement des Sciences|volume=10|pages=132–138|url=http://henripoincarepapers.univ-nantes.fr/chp/hp-pdf/hp1881af.pdf|archive-url=https://web.archive.org/web/20200801124731/http://henripoincarepapers.univ-nantes.fr/chp/hp-pdf/hp1881af.pdf|url-status=dead|archive-date=1 August 2020}}</ref><ref>{{Cite journal|author=Reynolds, W. F.|year=1993|title=Hyperbolic geometry on a hyperboloid|journal=The American Mathematical Monthly|volume=100|issue=5|pages=442–455|jstor=2324297|doi=10.1080/00029890.1993.11990430|s2cid=124088818 }}</ref> In addition, Poincaré's other models of hyperbolic geometry ([[Poincaré disk model]], [[Poincaré half-plane model]]) as well as the [[Beltrami–Klein model]] can be related to the relativistic velocity space (see [[Gyrovector space]]).

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

He discussed the "principle of relative motion" in two papers in 1900<ref name=action /><ref>{{Citation
| author=Poincaré, H. | year=1900 | title= Les relations entre la physique expérimentale et la physique mathématique | journal=Revue Générale des Sciences Pures et Appliquées | volume =11 | pages =1163–1175 | url=http://gallica.bnf.fr/ark:/12148/bpt6k17075r/f1167.table}}. Reprinted in "Science and Hypothesis", Ch. 9–10.</ref>
and named it the [[principle of relativity]] in 1904, according to which no physical experiment can discriminate between a state of uniform motion and a state of rest.<ref name=louis>{{Citation|author=Poincaré, Henri|year=1913|chapter=[[s:The Principles of Mathematical Physics|The Principles of Mathematical Physics]]|title=The Foundations of Science (The Value of Science)|pages=297–320|publisher=Science Press|place=New York|others= Article translated from 1904 original}} available in [https://books.google.com/books/about/The_Foundations_of_Science.html?id=mBvNabP35zoC&pg=PA297 online chapter from 1913 book]</ref>
In 1905 Poincaré wrote to Lorentz about Lorentz's paper of 1904, which Poincaré described as a "paper of supreme importance". In this letter he pointed out an error Lorentz had made when he had applied his transformation to one of Maxwell's equations, that for charge-occupied space, and also questioned the time dilation factor given by Lorentz.<ref name="univ-nantes">
{{Citation | author=Poincaré, H. | year=2007 | editor=Walter, S. A. | contribution= 38.3, Poincaré to H. A. Lorentz, May 1905 | title=La correspondance entre Henri Poincaré et les physiciens, chimistes, et ingénieurs |pages=255–257 |place=Basel | publisher=Birkhäuser|contribution-url=http://henripoincarepapers.univ-nantes.fr/chp/text/lorentz3.html}}</ref>
In a second letter to Lorentz, Poincaré gave his own reason why Lorentz's time dilation factor was indeed correct after all—it was necessary to make the Lorentz transformation form a group—and he gave what is now known as the relativistic velocity-addition law.<ref name="univ-nantes2">{{Citation | author=Poincaré, H. | year=2007 | editor=Walter, S. A. | contribution= 38.4, Poincaré to H. A. Lorentz, May 1905 | title=La correspondance entre Henri Poincaré et les physiciens, chimistes, et ingénieurs |pages=257–258 |place=Basel | publisher=Birkhäuser|contribution-url=http://henripoincarepapers.univ-nantes.fr/chp/text/lorentz4.html}}</ref>
Poincaré later delivered a paper at the meeting of the Academy of Sciences in Paris on 5 June 1905 in which these issues were addressed. In the published version of that he wrote:<ref name="1905 paper">[http://www.academie-sciences.fr/pdf/dossiers/Poincare/Poincare_pdf/Poincare_CR1905.pdf] (PDF) Membres de l'Académie des sciences depuis sa création : Henri Poincare. Sur la dynamique de l' electron. Note de H. Poincaré. C.R. T.140 (1905) 1504–1508.</ref>

<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>

and showed that the arbitrary function <math>\ell\left(\varepsilon\right)</math> must be unity for all <math>\varepsilon</math> (Lorentz had set <math>\ell = 1</math> by a different argument) to make the transformations form a group. In an enlarged version of the paper that appeared in 1906 Poincaré pointed out that the combination <math>x^2+ y^2+ z^2- c^2t^2</math> is [[Invariant (mathematics)|invariant]]. He noted that a Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing <math>ct\sqrt{-1}</math> as a fourth imaginary coordinate, and he used an early form of [[four-vector]]s.<ref name=long>{{Citation
| author=Poincaré, H. | year=1906 | title=Sur la dynamique de l'électron (On the Dynamics of the Electron) | journal=Rendiconti del Circolo Matematico Rendiconti del Circolo di Palermo | volume =21 | pages =129–176
| 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 name="Genesis 4-Vectors">{{cite book | last=Walter | first=Scott | title=The Genesis of General Relativity | chapter=Breaking in the 4-Vectors: The Four-Dimensional Movement in Gravitation, 1905–1910 |volume=3| publisher=Springer Netherlands |publication-place=Dordrecht | date=2007 | isbn=978-1-4020-3999-7 | doi=10.1007/978-1-4020-4000-9_18 | pages=1118–1178}}</ref> So it was [[Hermann Minkowski]] who worked out the consequences of this notion in 1907.<ref name="Genesis 4-Vectors"/><ref name="Raum und Zeit">{{cite journal | last = Minkowski | first = Hermann | date = September 1908 | title = Raum und Zeit | url = https://math.nyu.edu/~tschinke/papers/yuri/14minkowski/raum-und-zeit.pdf | journal = Jahresbericht der Deutschen Mathematiker-Vereinigung | volume = 18 | pages = 75–88 | access-date=11 May 2024 }}</ref>

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

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

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

{{blockquote|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. }}

In the above quote he refers to the Hertz assumption of total aether entrainment that was falsified by the [[Fizeau experiment]] but that experiment does indeed show that that light is partially "carried along" with a substance. Finally in 1908<ref name=poinc08>{{cite book 
|author=Poincaré, Henri
|year=1908–1913
|title=The foundations of science (Science and Method)
|chapter=[[s:The New Mechanics|The New Mechanics]]
|location=New York |publisher=Science Press
|pages=486–522}}</ref> he revisits the problem and ends with abandoning the principle of reaction altogether in favor of supporting a solution based in the inertia of aether itself.

{{blockquote|But we have seen above that Fizeau's experiment does not permit of our retaining the theory of Hertz; it is necessary therefore to adopt the theory of Lorentz, and consequently to renounce the principle of reaction. }}

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 [[Marie Curie]].

It was [[Albert Einstein]]'s concept of [[mass–energy equivalence]] (1905) that a body losing energy as radiation or heat was losing mass of amount ''m''&nbsp;=&nbsp;''E''/''c''<sup>2</sup> that resolved<ref name=darrigol>Darrigol 2005, Secondary sources on relativity</ref> Poincaré's paradox, without using any compensating mechanism within the ether.<ref>{{Citation|author=Einstein, A. |year=1905b |title=Ist die Trägheit eines Körpers von dessen Energieinhalt abhängig? |journal=Annalen der Physik |volume=18 |issue=13 |pages=639–643 |bibcode=1905AnP...323..639E |doi= 10.1002/andp.19053231314 |doi-access=free }}. 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 |s2cid=120361282 |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====
In 1905 Poincaré first proposed [[gravitational waves]] (''ondes gravifiques'') emanating from a body and propagating at the speed of light.  He wrote:
{{blockquote|It has become important to examine this hypothesis more closely and in particular to ask in what ways it would require us to modify the laws of gravitation.  That is what I have tried to determine;  at first I was led to assume that the propagation of gravitation is not instantaneous, but happens with the speed of light.<ref>"''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.''"</ref><ref name="1905 paper" />}}

====Poincaré and Einstein====
Einstein's first paper on relativity was published three months after Poincaré's short paper,<ref name="1905 paper" /> but before Poincaré's longer version.<ref name=long /> Einstein relied on the principle of relativity to derive the Lorentz transformations and used a similar clock synchronisation procedure ([[Einstein synchronisation]]) to the one that Poincaré (1900) had described, but Einstein's paper was remarkable in that it contained no references at all. Poincaré never acknowledged Einstein's work on [[special relativity]]. However, Einstein expressed sympathy with Poincaré's outlook obliquely in a letter to [[Hans Vaihinger]] on 3 May 1919, when Einstein considered Vaihinger's general outlook to be close to his own and Poincaré's to be close to Vaihinger's.<ref>{{cite book|series=The Collected Papers of Albert Einstein |url=http://einsteinpapers.press.princeton.edu/vol9-trans/52 |publisher=Princeton U.P. |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 | author-link = 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 titled "''Geometrie und Erfahrung'' (Geometry and Experience)" 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 recognized 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>

====Assessments on Poincaré and relativity====
{{Further|History of special relativity|Relativity priority dispute}}
Poincaré's work in the development of special relativity is well recognised,<ref name=darrigol /> though most historians stress that despite many similarities with Einstein's work, the two had very different research agendas and interpretations of the work.<ref>Galison 2003 and Kragh 1999, Secondary sources on relativity</ref> Poincaré developed a similar physical interpretation of local time and noticed the connection to signal velocity, but contrary to Einstein he continued to use the ether-concept in his papers and argued that clocks at rest in the ether show the "true" time, and moving clocks show the local time. So Poincaré tried to keep the relativity principle in accordance with classical concepts, while Einstein developed a mathematically equivalent kinematics based on the new physical concepts of the relativity of space and time.<ref>Holton (1988), 196–206</ref><ref name="Hentschel PhD">{{cite thesis |last=Hentschel|first=Klaus |date=1990 |title=Interpretationen und Fehlinterpretationen der speziellen und der allgemeinen Relativitätstheorie durch Zeitgenossen Albert Einsteins |degree=PhD |publisher=University of Hamburg|pages=3–13}}</ref><ref>Miller (1981), 216–217</ref><ref>Darrigol (2005), 15–18</ref><ref>Katzir (2005), 286–288</ref>

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

===Algebra and number theory===
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><ref name="Poincaré Lorentz l’´electron"> {{cite journal | last = Poincaré | first = Henri | date = 1905 | title = Sur la dynamique de l'électron | journal = Comptes rendus des séances de l'Académie des Sciences | volume = 140 | pages = 1504–1508 }}</ref> He also made major contributions to the theory of discrete groups and their representations.
[[Image:Mug and Torus morph.gif|{{center|Topological transformation of a mug into a torus }}|thumb|190x190px]]
[[File:Poincaré-7.jpg|alt=Title page to volume I of Les Méthodes Nouvelles de la Mécanique Céleste (1892)|thumb|192x192px|Title page to volume I of ''Les Méthodes Nouvelles de la Mécanique Céleste'' (1892)]]

===Topology===
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|pp=419–435}}

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 number]]s and the [[fundamental group]]. Poincaré proved a formula relating the number of edges, [[Triangulated irregular network|vertices]] and faces of ''n''-dimensional [[polyhedron]] (the [[Euler characteristic|Euler–Poincaré theorem]]) and gave the first precise formulation of the intuitive notion of dimension.<ref name="Aleksandrov Poincaré ">{{cite journal | last=Aleksandrov | first=P S | title=Poincaré and topology | journal=Russian Mathematical Surveys | volume=27 | issue=1 | date=28 February 1972 | issn=0036-0279 | doi=10.1070/RM1972v027n01ABEH001365 | pages=157–168| bibcode=1972RuMaS..27..157A }}</ref>

===Astronomy and celestial mechanics===
[[File:N-body problem (3).gif|frame|left|{{center| Chaotic motion in three-body problem (computer simulation)}}]]
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 [[algebra]]ic 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 p. 254]</ref>

These monographs include an idea of Poincaré, which later became the basis for mathematical "[[chaos theory]]" (see, in particular, the [[Poincaré recurrence theorem]]) and the general theory of [[dynamical system]]s.
Poincaré authored important works on [[astronomy]] for the [[Hydrostatic equilibrium|equilibrium figures of a gravitating rotating fluid]]. He introduced the important concept of [[Bifurcation theory|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 name="Gold Medal to Poincaré ">{{cite journal |author-last=Darwin | author-first=G.H.| title=Address Delivered by the President, Professor G. H. Darwin, on presenting the Gold Medal of the Society to M. H. Poincaré | journal=Monthly Notices of the Royal Astronomical Society | volume=60 | issue=5 | date=1900 | issn=0035-8711 | doi=10.1093/mnras/60.5.406 | pages=406–416| doi-access=free}}</ref>

===Differential equations and mathematical physics===
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 point|saddle]], [[Focus (geometry)|focus]], [[Center (algebra)|center]], [[Vertex (graph theory)|node]]), introduced the concept of a [[limit cycle]] and the [[Control flow#Loop system cross-reference table|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|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| publisher=Springer }}</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
 File: Phase Portrait Stable Node.svg | Node
</gallery>