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Poincaré, Henri

Henri Poincaré
Jules Henri Poincaré (1854–1912). Photograph from the frontispiece of the 1913 edition of Last Thoughts.
Born	29 April 1854(1854-04-29) Nancy, Meurthe-et-Moselle
Died	17 July 1912(1912-07-17) (aged 58) Paris
Residence	France
Nationality	French
Fields	Mathematician and physicist
Institutions	Corps des Mines Caen University La Sorbonne Bureau des Longitudes
Alma mater	Lycée Nancy École Polytechnique École des Mines
Doctoral advisor	Charles Hermite
Doctoral students	Louis Bachelier Dimitrie Pompeiu Mihailo Petrović
Other notable students	Tobias Dantzig
Known for	Poincaré conjecture Three-body problem Topology Special relativity Poincaré–Hopf theorem Poincaré duality Poincaré–Birkhoff–Witt theorem Poincaré inequality Hilbert–Poincaré series Poincaré metric Rotation number Coining term 'Betti number' Chaos theory Sphere-world Poincaré–Bendixson theorem Poincaré–Lindstedt method Poincaré recurrence theorem
Influences	Lazarus Fuchs
Influenced	Louis Rougier George David Birkhoff
Notable awards	RAS Gold Medal (1900) Sylvester Medal (1901) Matteucci Medal (1905) Bolyai Prize (1905) Bruce Medal (1911)
Signature
Notes He was a cousin of Pierre Boutroux.

Jules Henri Poincaré (29 April 1854 – 17 July 1912) (French pronunciation: [ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe])^[1] was a French mathematician, theoretical physicist, and a philosopher of science. He is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime.

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

Poincaré introduced the modern principle of relativity and was the first to present the Lorentz transformations in their modern symmetrical form. Poincaré discovered the remaining relativistic velocity transformations and recorded them in a letter to Dutch physicist Hendrik Lorentz (1853–1928) in 1905. Thus he obtained perfect invariance of all of Maxwell's equations, an important step in the formulation of the theory of special relativity.

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

[edit] Life

Poincaré was born on 29 April 1854 in Cité Ducale neighborhood, Nancy, Meurthe-et-Moselle into an influential family (Belliver, 1956). His father Leon Poincaré (1828–1892) was a professor of medicine at the University of Nancy (Sagaret, 1911). His adored younger sister Aline married the spiritual philosopher Emile Boutroux. Another notable member of Jules' family was his cousin, Raymond Poincaré, who would become the President of France, 1913 to 1920, and a fellow member of the Académie française.^[2]

[edit] Education

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

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

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

Poincaré entered the École Polytechnique in 1873. There he studied mathematics as a student of Charles Hermite, continuing to excel and publishing his first paper (Démonstration nouvelle des propriétés de l'indicatrice d'une surface) in 1874. He graduated in 1875 or 1876. He went on to study at the École des Mines, continuing to study mathematics in addition to the mining engineering syllabus and received the degree of ordinary engineer in March 1879.

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

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

The young Henri Poincaré

[edit] Career

Soon after, he was offered a post as junior lecturer in mathematics at Caen University, but he never fully abandoned his mining career to mathematics. He worked at the Ministry of Public Services as an engineer in charge of northern railway development from 1881 to 1885. He eventually became chief engineer of the Corps de Mines in 1893 and inspector general in 1910.

Beginning in 1881 and for the rest of his career, he taught at the University of Paris (the Sorbonne). He was initially appointed as the maître de conférences d'analyse (associate professor of analysis) (Sageret, 1911). Eventually, he held the chairs of Physical and Experimental Mechanics, Mathematical Physics and Theory of Probability, and Celestial Mechanics and Astronomy.

Also in that same year, Poincaré married Miss Poulain d'Andecy. Together they had four children: Jeanne (born 1887), Yvonne (born 1889), Henriette (born 1891), and Léon (born 1893).

In 1887, at the young age of 32, Poincaré was elected to the French Academy of Sciences. He became its president in 1906, and was elected to the Académie française in 1909.

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

The Poincaré family grave at the Cimetière du Montparnasse

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

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

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

A former French Minister of Education, Claude Allègre, has recently (2004) proposed that Poincaré be reburied in the Panthéon in Paris, which is reserved for French citizens only of the highest honour.^[3]

[edit] Students

Poincaré had two notable doctoral students at the University of Paris, Louis Bachelier (1900) and Dimitrie Pompeiu (1905).^[4]

[edit] Work

[edit] Summary

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

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

Among the specific topics he contributed to are the following:

algebraic topology
the theory of analytic functions of several complex variables
the theory of abelian functions
algebraic geometry
Poincaré was responsible for formulating one of the most famous problems in mathematics. Known as the Poincaré conjecture, it is a problem in topology.
Poincaré recurrence theorem
hyperbolic geometry
number theory
the three-body problem
the theory of diophantine equations
the theory of electromagnetism
the special theory of relativity
In an 1894 paper, he introduced the concept of 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 sphere and the Poincaré map.
Poincaré on "everybody's belief" in the 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.^[5]^[6]

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

“

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

”

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^[7]). The version finally printed contained many important ideas which lead to the theory of chaos. The problem as stated originally was finally solved by Karl F. Sundman for n = 3 in 1912 and was generalised to the case of n > 3 bodies by Qiudong Wang in the 1990s.

[edit] Work on relativity

Marie Curie and Poincaré talk at the 1911 Solvay Conference.

Main articles: Lorentz ether theory and History of special relativity

[edit] 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" $t^\prime = t-v x/c^2 \,$ ^[8] 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).^[9] Poincaré was a constant interpreter (and sometimes friendly critic) of Lorentz's theory. Poincaré as a philosopher was interested in the "deeper meaning". Thus he interpreted Lorentz's theory and in so doing he came up with many insights that are now associated with special relativity. In The Measure of Time (1898), Poincaré said, " 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.^[10] 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.^[11]

Wikisource has original works written by or about: Henri Poincaré

[edit] Principle of relativity and Lorentz transformations

He discussed the "principle of relative motion" in two papers in 1900^[11]^[12] 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.^[13] 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.^[14] 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 gave what is now known as the relativistic velocity-addition law.^[15] 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:^[16]

“

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:

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

”

and showed that the arbitrary function $\ell\left(\varepsilon\right)$ must be unity for all $\varepsilon$ (Lorentz had set $\ell = 1$ 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 $x 2 + y 2 + z 2 - c 2 t 2$ is invariant. He noted that a Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing $ct\sqrt{-1}$ as a fourth imaginary coordinate, and he used an early form of four-vectors.^[17] Poincaré’s attempt at a four-dimensional reformulation of the new mechanics was rejected by himself 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.^[18] So it was Hermann Minkowski who worked out the consequences of this notion in 1907.

[edit] Mass–energy relation

Like others before, Poincaré (1900) discovered a relation between mass and electromagnetic energy. While studying the conflict between the action/reaction principle and Lorentz ether theory, he tried to determine whether the center of gravity still moves with a uniform velocity when electromagnetic fields are included.^[11] He noticed that the action/reaction principle does not hold for matter alone, but that the electromagnetic field has its own momentum. Poincaré concluded that the electromagnetic field energy of an electromagnetic wave behaves like a fictitious fluid ("fluide fictif") with a mass density of E/c². If the center of mass frame is defined by both the mass of matter and the mass of the fictitious fluid, and if the fictitious fluid is indestructible — it's neither created or destroyed — then the motion of the center of mass frame remains uniform. But electromagnetic energy can be converted into other forms of energy. So Poincaré assumed that there exists a non-electric energy fluid at each point of space, into which electromagnetic energy can be transformed and which also carries a mass proportional to the energy. In this way, the motion of the center of mass remains uniform. Poincaré said that one should not be too surprised by these assumptions, since they are only mathematical fictions.

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

Poincaré himself came back to this topic in his St. Louis lecture (1904).^[13] This time (and later also in 1908) he rejected^[19] the possibility that energy carries mass and also the possibility, that motions in the ether can compensate the above mentioned problems:

“

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

”

He also discussed two other unexplained effects: (1) non-conservation of mass implied by Lorentz's variable mass $γ m$ , 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.

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^[20] Poincare's paradox, without using any compensating mechanism within the ether.^[21] 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.^[22]

[edit] Poincaré and Einstein

Einstein's first paper on relativity was published three months after Poincaré's short paper,^[16] but before Poincaré's longer version.^[17] It relied on the principle of relativity to derive the Lorentz transformations and used a similar clock synchronisation procedure (Einstein synchronisation) that Poincaré (1900) had described, but was remarkable in that it contained no references at all. Poincaré never acknowledged Einstein's work on special relativity. Einstein acknowledged Poincaré 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 ...."^[23]

[edit] Assessments

Further information: History of special relativity and Relativity priority dispute

Poincaré's work in the development of special relativity is well recognised,^[20] though most historians stress that despite many similarities with Einstein's work, the two had very different research agendas and interpretations of the work.^[24] 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 in the ether show the "true" time, and moving clocks show the local time. So Poincaré tried to bring the relativity principle in accordance with classical physics, while Einstein developed a mathematically equivalent kinematics based on the new physical concepts of the relativity of space and time.^[25]^[26]^[27]^[28]^[29] 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.^[30]

[edit] Character

Photographic portrait of H. Poincaré by Henri Manuel

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.

The mathematician Darboux claimed he was un intuitif (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. He believed that logic was not a way to invent but a way to structure ideas and that logic limits ideas.

[edit] Toulouse' characterisation

Poincaré's mental organisation was not only interesting to Poincaré himself but also to Toulouse, a psychologist of the Psychology Laboratory of the School of Higher Studies in Paris. Toulouse wrote a book entitled Henri Poincaré (1910).^[31] In it, he discussed Poincaré's regular schedule:

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

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

He was ambidextrous and nearsighted.

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

These abilities were offset to some extent by his shortcomings:

He was physically clumsy and artistically inept.

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

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

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

His method of thinking is well summarised as:

"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.") Belliver (1956)

[edit] Attitude towards transfinite numbers

Poincaré was dismayed by Georg Cantor's theory of transfinite numbers, and referred to it as a "disease" from which mathematics would eventually be cured.^[32] Poincare said, "There is no actual infinite; the Cantorians have forgotten this, and that is why they have fallen into contradiction."^[33]

[edit] View on economics

Poincaré saw mathematical work in economics and finance as peripheral. In 1900 Poincaré commented on Louis Bachelier's thesis "The Theory of Speculation", saying: "M. Bachelier has evidenced an original and precise mind [but] the subject is somewhat remote from those our other candidates are in the habit of treating." (Bernstein, 1996, pp. 199–200) Bachelier's work explained what was then the French government's pricing options on French Bonds and anticipated many of the pricing theories in financial markets used today.^[34]

[edit] Honours

Awards

Oscar II, King of Sweden's mathematical competition (1887)
American Philosophical Society 1899
Gold Medal of the Royal Astronomical Society of London (1900)
Bolyai Prize in 1905
Matteucci Medal 1905
French Academy of Sciences 1906
Académie Française 1909
Bruce Medal (1911)

Named after him

Poincaré Prize (Mathematical Physics International Prize)
Annales Henri Poincaré (Scientific Journal)
Poincaré Seminar (nicknamed "Bourbaphy")
The crater Poincaré on the Moon
Asteroid 2021 Poincaré

[edit] Philosophy

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

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.

Poincaré believed that arithmetic is a synthetic science. He argued that Peano's axioms canno

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