Henri Poincaré
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Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
,
theoretical physicist Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experime ...
,
engineer Engineers, as practitioners of engineering, are professionals who invent, design, analyze, build and test machines, complex systems, structures, gadgets and materials to fulfill functional objectives and requirements while considering the limit ...
, and
philosopher of science A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek th ...
. He is often described as a
polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose knowledge spans a substantial number of subjects, known to draw on complex bodies of knowledge to solve specific pro ...
, 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 A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...
, he made many original fundamental contributions to
pure Pure may refer to: Computing * A pure function * A pure virtual function * PureSystems, a family of computer systems introduced by IBM in 2012 * Pure Software, a company founded in 1991 by Reed Hastings to support the Purify tool * Pure-FTPd, F ...
and
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
,
mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...
, and
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
. In his research on the
three-body problem In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...
, Poincaré became the first person to discover a chaotic
deterministic system In mathematics, computer science and physics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given sta ...
which laid the foundations of modern chaos theory. He is also considered to be one of the founders of the field of
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. Poincaré made clear the importance of paying attention to the invariance of laws of physics under different transformations, and was the first to present the
Lorentz transformations In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
in their modern symmetrical form. Poincaré discovered the remaining relativistic velocity transformations and recorded them in a letter to Hendrik Lorentz in 1905. Thus he obtained perfect invariance of all of
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
, an important step in the formulation of the theory of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws ...
. In 1905, Poincaré first proposed gravitational waves (''ondes gravifiques'') emanating from a body and propagating at the speed of light as being required by the Lorentz transformations. The
Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
used in physics and mathematics was named after him. Early in the 20th century he formulated the
Poincaré conjecture In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured ...
that became over time one of the famous
unsolved problems in mathematics Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Eucli ...
until it was solved in 2002–2003 by
Grigori Perelman Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...
.


Life

Poincaré was born on 29 April 1854 in Cité Ducale neighborhood,
Nancy, Meurthe-et-Moselle Nancy ; Lorraine Franconian: ''Nanzisch'' is the prefecture of the northeastern French department of Meurthe-et-Moselle. It was the capital of the Duchy of Lorraine, which was annexed by France under King Louis XV in 1766 and replaced by a provi ...
, into an influential French family. His father Léon Poincaré (1828–1892) was a professor of medicine at the
University of Nancy A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States, th ...
. His younger sister Aline married the spiritual philosopher
Émile Boutroux Étienne Émile Marie Boutroux (; 28 July 1845 – 22 November 1921) was an eminent 19th-century French philosopher of science and religion, and a historian of philosophy. He was a firm opponent of materialism in science. He was a spiritual p ...
. Another notable member of Henri's family was his cousin, Raymond Poincaré, a fellow member of the Académie française, who was President of France from 1913 to 1920.The Internet Encyclopedia of Philosophy
Jules Henri Poincaré article by Mauro Murzi – Retrieved November 2006.


Education

During his childhood he was seriously ill for a time with
diphtheria Diphtheria is an infection caused by the bacterium '' Corynebacterium diphtheriae''. Most infections are asymptomatic or have a mild clinical course, but in some outbreaks more than 10% of those diagnosed with the disease may die. Signs and s ...
and received special instruction from his mother, Eugénie Launois (1830–1897). In 1862, Henri entered the Lycée in Nancy (now renamed the in his honour, along with
Henri Poincaré University The Henri Poincaré University, or Nancy 1, nicknamed UHP, was a public research university located in Nancy, France. UHP was merged into University of Lorraine in 2012, and was previously a member of the Nancy-Université federation, belongin ...
, also in Nancy). He spent eleven years at the Lycée and during this time he proved to be one of the top students in every topic he studied. He excelled in written composition. His mathematics teacher described him as a "monster of mathematics" and he won first prizes in the
concours général In France, the Concours Général is the most prestigious academic competition held every year between students of ''Première'' (11th grade) and ''Terminale'' (12th and final grade) in almost all subjects taught in both general, technological and ...
, a competition between the top pupils from all the Lycées across France. His poorest subjects were music and physical education, where he was described as "average at best". However, poor eyesight and a tendency towards absentmindedness may explain these difficulties. He graduated from the Lycée in 1871 with a baccalauréat in both letters and sciences. During the Franco-Prussian War of 1870, he served alongside his father in the
Ambulance Corps Emergency medical services (EMS), also known as ambulance services or paramedic services, are emergency services that provide urgent pre-hospital treatment and stabilisation for serious illness and injuries and transport to definitive care. ...
. Poincaré entered the
École Polytechnique École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing in région Île-de-France * École, Savoi ...
as the top qualifier in 1873 and graduated in 1875. There he studied mathematics as a student of
Charles Hermite Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. Hermi ...
, continuing to excel and publishing his first paper (''Démonstration nouvelle des propriétés de l'indicatrice d'une surface'') in 1874. From November 1875 to June 1878 he studied at the
École des Mines École may refer to: * an elementary school in the French educational stages normally followed by secondary education Secondary education or post-primary education covers two phases on the International Standard Classification of Education scal ...
, while continuing the study of mathematics in addition to the
mining engineering Mining in the engineering discipline is the extraction of minerals from underneath, open pit, above or on the ground. Mining engineering is associated with many other disciplines, such as mineral processing, exploration, excavation, geology, and ...
syllabus, and received the degree of ordinary mining engineer in March 1879. As a graduate of the École des Mines, he joined the
Corps des Mines The ''Corps des mines'' is the foremost technical Grand Corps of the French State (grands corps de l'Etat). It is composed of the state industrial engineers. The Corps is attached to the French Ministry of Economy and Finance. Its purpose is to e ...
as an inspector for the
Vesoul Vesoul () is a commune in the Haute-Saône department in the region of Bourgogne-Franche-Comté located in eastern France. It is the most populated municipality of the department with inhabitants in 2014. The same year, the Communauté d'agglo ...
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 Science in mathematics under the supervision of Charles Hermite. His doctoral thesis was in the field of differential equations. It was named ''Sur les propriétés des fonctions définies par les équations aux différences partielles''. Poincaré devised a new way of studying the properties of these equations. He not only faced the question of determining the integral of such equations, but also was the first person to study their general geometric properties. He realised that they could be used to model the behaviour of multiple bodies in free motion within the
Solar System The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...
. Poincaré graduated from the
University of Paris , image_name = Coat of arms of the University of Paris.svg , image_size = 150px , caption = Coat of Arms , latin_name = Universitas magistrorum et scholarium Parisiensis , motto = ''Hic et ubique terrarum'' (Latin) , mottoeng = Here and a ...
in 1879.


First scientific achievements

After receiving his degree, Poincaré began teaching as junior lecturer in mathematics at the
University of Caen The University of Caen Normandy (French: ''Université de Caen Normandie''), also known as Unicaen, is a public university in Caen, France. History The institution was founded in 1432 by John of Lancaster, 1st Duke of Bedford, the first rector ...
in Normandy (in December 1879). At the same time he published his first major article concerning the treatment of a class of
automorphic function In mathematics, an automorphic function is a function on a space that is invariant under the action of some group, in other words a function on the quotient space. Often the space is a complex manifold and the group is a discrete group. Factor ...
s. There, in Caen, he met his future wife, Louise Poulain d'Andecy (1857–1934), granddaughter of
Isidore Geoffroy Saint-Hilaire Isidore Geoffroy Saint-Hilaire (16 December 1805 – 10 November 1861) was a French zoology, zoologist and an authority on deviation from normal structure. In 1854 he coined the term ''éthologie'' (ethology). Biography He was born in Paris ...
and great-granddaughter of
Étienne Geoffroy Saint-Hilaire Étienne Geoffroy Saint-Hilaire (15 April 177219 June 1844) was a French naturalist who established the principle of "unity of composition". He was a colleague of Jean-Baptiste Lamarck and expanded and defended Lamarck's evolutionary theories ...
and on 20 April 1881, they married. Together they had four children: Jeanne (born 1887), Yvonne (born 1889), Henriette (born 1891), and Léon (born 1893). Poincaré immediately established himself among the greatest mathematicians of Europe, attracting the attention of many prominent mathematicians. In 1881 Poincaré was invited to take a teaching position at the Faculty of Sciences of the
University of Paris , image_name = Coat of arms of the University of Paris.svg , image_size = 150px , caption = Coat of Arms , latin_name = Universitas magistrorum et scholarium Parisiensis , motto = ''Hic et ubique terrarum'' (Latin) , mottoeng = Here and a ...
; he accepted the invitation. During the years 1883 to 1897, he taught
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
in the
École Polytechnique École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing in région Île-de-France * École, Savoi ...
. In 1881–1882, Poincaré created a new branch of mathematics:
qualitative theory of differential equations In mathematics, the qualitative theory of differential equations studies the behavior of differential equations by means other than finding their solutions. It originated from the works of Henri Poincaré and Aleksandr Lyapunov. There are relatively ...
. He showed how it is possible to derive the most important information about the behavior of a family of solutions without having to solve the equation (since this may not always be possible). He successfully used this approach to problems in
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
and
mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...
.


Career

He never fully abandoned his career in the mining administration 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 des Mines The ''Corps des mines'' is the foremost technical Grand Corps of the French State (grands corps de l'Etat). It is composed of the state industrial engineers. The Corps is attached to the French Ministry of Economy and Finance. Its purpose is to e ...
in 1893 and inspector general in 1910. Beginning in 1881 and for the rest of his career, he taught at the
University of Paris , image_name = Coat of arms of the University of Paris.svg , image_size = 150px , caption = Coat of Arms , latin_name = Universitas magistrorum et scholarium Parisiensis , motto = ''Hic et ubique terrarum'' (Latin) , mottoeng = Here and a ...
(the
Sorbonne Sorbonne may refer to: * Sorbonne (building), historic building in Paris, which housed the University of Paris and is now shared among multiple universities. *the University of Paris (c. 1150 – 1970) *one of its components or linked institution, ...
). He was initially appointed as the ''maître de conférences d'analyse'' (associate professor of analysis). Eventually, he held the chairs of Physical and Experimental Mechanics, Mathematical Physics and Theory of Probability, and Celestial Mechanics and Astronomy. In 1887, at the young age of 32, Poincaré was elected to the French Academy of Sciences. He became its president in 1906, and was elected to the Académie française on 5 March 1908. In 1887, he won Oscar II, King of Sweden's mathematical competition for a resolution of the
three-body problem In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...
concerning the free motion of multiple orbiting bodies. (See
three-body problem In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...
section below.) In 1893, Poincaré joined the French Bureau des Longitudes, which engaged him in the synchronisation of time around the world. In 1897 Poincaré backed an unsuccessful proposal for the decimalisation of circular measure, and hence time and
longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek lette ...
. 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 1904, he intervened in the
trials In law, a trial is a coming together of parties to a dispute, to present information (in the form of evidence) in a tribunal, a formal setting with the authority to adjudicate claims or disputes. One form of tribunal is a court. The tribun ...
of Alfred Dreyfus, attacking the spurious scientific claims regarding evidence brought against Dreyfus. Poincaré was the President of the Société Astronomique de France (SAF), the French astronomical society, from 1901 to 1903.


Students

Poincaré had two notable doctoral students at the University of Paris,
Louis Bachelier Louis Jean-Baptiste Alphonse Bachelier (; 11 March 1870 – 28 April 1946) was a French mathematician at the turn of the 20th century. He is credited with being the first person to model the stochastic process now called Brownian motion, as part ...
(1900) and
Dimitrie Pompeiu Dimitrie D. Pompeiu (; – 8 October 1954) was a Romanian mathematician, professor at the University of Bucharest, titular member of the Romanian Academy, and President of the Chamber of Deputies. Biography He was born in 1873 in Broscăuți, ...
(1905).


Death

In 1912, Poincaré underwent surgery for a
prostate The prostate is both an accessory gland of the male reproductive system and a muscle-driven mechanical switch between urination and ejaculation. It is found only in some mammals. It differs between species anatomically, chemically, and phys ...
problem and subsequently died from an
embolism An embolism is the lodging of an embolus, a blockage-causing piece of material, inside a blood vessel. The embolus may be a blood clot (thrombus), a fat globule (fat embolism), a bubble of air or other gas ( gas embolism), amniotic fluid (am ...
on 17 July 1912, in Paris. He was 58 years of age. He is buried in the Poincaré family vault in the Cemetery of Montparnasse, Paris, in section 16 close to the gate Rue Émile-Richard. A former French Minister of Education,
Claude Allègre Claude Allègre (; born 31 March 1937) is a French politician and scientist. Scientific work The main scientific area of Claude Allègre was geochemistry. Allègre co-authored an ''Introduction to geochemistry'' in 1974. Since the 1980s, he ...
, proposed in 2004 that Poincaré be reburied in the
Panthéon The Panthéon (, from the Classical Greek word , , ' empleto all the gods') is a monument in the 5th arrondissement of Paris, France. It stands in the Latin Quarter, atop the , in the centre of the , which was named after it. The edifice was b ...
in Paris, which is reserved for French citizens of the highest honour.


Work


Summary

Poincaré made many contributions to different fields of pure and applied mathematics such as:
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
,
fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
,
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
,
electricity Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as describ ...
,
telegraphy Telegraphy is the long-distance transmission of messages where the sender uses symbolic codes, known to the recipient, rather than a physical exchange of an object bearing the message. Thus flag semaphore is a method of telegraphy, whereas ...
,
capillarity Capillary action (sometimes called capillarity, capillary motion, capillary rise, capillary effect, or wicking) is the process of a liquid flowing in a narrow space without the assistance of, or even in opposition to, any external forces li ...
, elasticity,
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of th ...
,
potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...
,
quantum theory Quantum theory may refer to: Science *Quantum mechanics, a major field of physics *Old quantum theory, predating modern quantum mechanics * Quantum field theory, an area of quantum mechanics that includes: ** Quantum electrodynamics ** Quantum ...
,
theory of relativity The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...
and
physical cosmology Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of f ...
. 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 Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
(a field that Poincaré virtually invented) * the theory of analytic functions of several complex variables * the theory of abelian functions * algebraic geometry *the
Poincaré conjecture In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured ...
, proven in 2003 by
Grigori Perelman Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...
. * Poincaré recurrence theorem *
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...
*
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
*the
three-body problem In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...
* the theory of diophantine equations *
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
* the special theory of relativity *the fundamental group *In the field of differential equations Poincaré has given many results that are critical for the qualitative theory of differential equations, for example the
Poincaré sphere Poincaré sphere may refer to: * Poincaré sphere (optics), a graphical tool for visualizing different types of polarized light ** Bloch sphere, a related tool for representing states of a two-level quantum mechanical system * Poincaré homology s ...
and the
Poincaré map In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower-dimensiona ...
. *Poincaré on "everybody's belief" in the ''Normal Law of Errors'' (see
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
for an account of that "law") *Published an influential paper providing a novel mathematical argument in support of quantum mechanics.


Three-body problem

The problem of finding the general solution to the motion of more than two orbiting bodies in the
Solar System The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...
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, 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 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.
In case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was finally awarded to Poincaré, even though he did not solve the original problem. One of the judges, the distinguished Karl Weierstrass, said, ''"This work cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that its publication will inaugurate a new era in the history of celestial mechanics."'' (The first version of his contribution even contained a serious error; for details see the article by Diacu and the book by June Barrow-Green, Barrow-Green). The version finally printed 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'' = 3 in 1912 and was generalised to the case of ''n'' > 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.


Work on relativity


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 \, and introduced the hypothesis of length contraction to explain the failure of optical and electrical experiments to detect motion relative to the aether (see Michelson–Morley experiment). Poincaré was a constant interpreter (and sometimes friendly critic) of Lorentz's theory. Poincaré as a philosopher was interested in the "deeper meaning". Thus he interpreted Lorentz's theory and in so doing he came up with many insights that are now associated with special relativity. In 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. 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.. See also th
English translation
/ref>


Principle of relativity and Lorentz transformations

In 1881 Poincaré described
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...
in terms of the hyperboloid model, formulating transformations leaving invariant the Lorentz interval x^2+y^2-z^2=-1, which makes them mathematically equivalent to the Lorentz transformations in 2+1 dimensions. In addition, Poincaré's other models of hyperbolic geometry (Poincaré disk model, Poincaré half-plane model) as well as the Beltrami–Klein model can be related to the relativistic velocity space (see Gyrovector space). In 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. 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. He discussed the "principle of relative motion" in two papers in 1900 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. available i
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. 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. 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:
(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.
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.
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^2t^2 is Invariant (mathematics), invariant. He noted that a Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing ct\sqrt as a fourth imaginary coordinate, and he used an early form of four-vectors. (Wikisource translation) Poincaré expressed a lack of interest in a four-dimensional reformulation of his new mechanics in 1907, because in his opinion the translation of physics into the language of four-dimensional geometry would entail too much effort for limited profit. So it was Hermann Minkowski who worked out the consequences of this notion in 1907.


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. 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—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). He rejected the possibility that energy carries mass and criticized his own solution to compensate the above-mentioned problems: 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 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. He also discussed two other unexplained effects: (1) non-conservation of mass implied by Lorentz's variable mass \gamma m, 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'' = ''E''/''c''2 that resolvedDarrigol 2005, Secondary sources on relativity 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.


Gravitational waves

In 1905 Poincaré first proposed gravitational waves (''ondes gravifiques'') emanating from a body and propagating at the speed of light. He wrote:


Poincaré and Einstein

Einstein's first paper on relativity was published three months after Poincaré's short paper, but before Poincaré's longer version. Einstein relied on the principle of relativity to derive the Lorentz transformations and used a similar clock synchronisation procedure (Einstein synchronisation) to the one that Poincaré (1900) had described, but Einstein's paper was remarkable in that it contained no references at all. Poincaré never acknowledged Einstein's work on
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws ...
. However, Einstein expressed sympathy with Poincaré's outlook obliquely in a letter to Hans Vaihinger on 3 May 1919, when Einstein considered Vaihinger's general outlook to be close to his own and Poincaré's to be close to Vaihinger's. In public, Einstein acknowledged Poincaré posthumously in the text of a lecture in 1921 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 ...."


Assessments on Poincaré and relativity

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


Algebra and number theory

Poincaré introduced group theory to physics, and was the first to study the group of
Lorentz transformations In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
. He also made major contributions to the theory of discrete groups and their representations.


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. 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, 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.


Astronomy and celestial mechanics

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. 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. 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).


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). In these articles, he built a new branch of mathematics, called "
qualitative theory of differential equations In mathematics, the qualitative theory of differential equations studies the behavior of differential equations by means other than finding their solutions. It originated from the works of Henri Poincaré and Aleksandr Lyapunov. There are relatively ...
". 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 Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...
and
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
, and the methods used were the basis of its topological works. 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


Character

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 Thought, thinking to how he made several discoveries. The mathematician Jean Gaston Darboux, Darboux claimed he was ''un intuitif'' (an intuitive), arguing that this is demonstrated by the fact that he worked so often by visual representation. Jacques Hadamard wrote that Poincaré's research demonstrated marvelous clarity and Poincaré himself wrote that he believed that logic was not a way to invent but a way to structure ideas and that logic limits ideas.


Toulouse's characterisation

Poincaré's mental organisation was interesting not only to Poincaré himself but also to Édouard Toulouse, a psychologist of the Psychology Laboratory of the School of Higher Studies in Paris. Toulouse wrote a book entitled ''Henri Poincaré'' (1910). In it, he discussed Poincaré's regular schedule: * He worked during the same times each day in short periods of time. He undertook mathematical research for four hours a day, between 10 a.m. and noon then again from 5 p.m. to 7 p.m.. He would read articles in journals later in the evening. * 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 Human multitasking, 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:


Publications

* * * * * * * * * * *


Honours

Awards *Oscar II, King of Sweden's mathematical competition (1887) *Foreign member of the Royal Netherlands Academy of Arts and Sciences (1897) *American Philosophical Society 1899 *Gold Medal of the Royal Astronomical Society of London (1900) *Bolyai Prize in 1905 *Matteucci Medal 1905 * French Academy of Sciences 1906 * Académie française 1909 *Bruce Medal (1911) Named after him *Institut Henri Poincaré (mathematics and theoretical physics center) *Poincaré Prize (Mathematical Physics International Prize) *Annales Henri Poincaré (Scientific Journal) *Poincaré Seminar (nicknamed "Bourbaphy") *The crater Poincaré (crater), Poincaré on the Moon *Asteroid 2021 Poincaré *List of things named after Henri Poincaré Henri Poincaré did not receive the Nobel Prize in Physics, but he had influential advocates like Henri Becquerel or committee member Gösta Mittag-Leffler. The nomination archive reveals that Poincaré received a total of 51 nominations between 1904 and 1912, the year of his death. Of the 58 nominations for the 1910 Nobel Prize, 34 named Poincaré. Nominators included Nobel laureates Hendrik Lorentz and Pieter Zeeman (both of 1902), Marie Curie (of 1903), Albert Michelson (of 1907), Gabriel Lippmann (of 1908) and Guglielmo Marconi (of 1909). The fact that renowned Theoretical physics, theoretical physicists like Poincaré, Ludwig Boltzmann, Boltzmann or Josiah Willard Gibbs, Gibbs were not awarded the Nobel Prize in Physics, Nobel Prize is seen as evidence that the Nobel committee had more regard for experimentation than theory. In Poincaré's case, several of those who nominated him pointed out that the greatest problem was to name a specific discovery, invention, or technique.


Philosophy

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 1902 book ''Science and Hypothesis'': Poincaré believed that arithmetic is Analytic/synthetic distinction, synthetic. He argued that Peano's axioms cannot be proven non-circularly with the Inductive reasoning, principle of induction (Murzi, 1998), therefore concluding that arithmetic is ''A priori and a posteriori, a priori'' Analytic–synthetic distinction, 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. However, Poincaré did not share Kantianism, 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". 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 (mathematics), 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.


Free will

Poincaré's famous lectures before the Société de Psychologie in Paris (published as ''Science and Hypothesis'', ''The Value of Science'', and ''Science and Method'') were cited by Jacques Hadamard as the source for the idea that creativity and Innovation, invention consist of two mental stages, first random combinations of possible solutions to a problem, followed by a critical thinking, critical evaluation. Although he most often spoke of a deterministic universe, Poincaré said that the subconscious generation of new possibilities involves Randomness, chance.
It is certain that the combinations which present themselves to the mind in a kind of sudden illumination after a somewhat prolonged period of unconscious work are generally useful and fruitful combinations... all the combinations are formed as a result of the automatic action of the subliminal ego, but those only which are interesting find their way into the field of consciousness... A few only are harmonious, and consequently at once useful and beautiful, and they will be capable of affecting the geometrician's special sensibility I have been speaking of; which, once aroused, will direct our attention upon them, and will thus give them the opportunity of becoming conscious... In the subliminal ego, on the contrary, there reigns what I would call liberty, if one could give this name to the mere absence of discipline and to disorder born of chance.
Poincaré's two stages—random combinations followed by selection—became the basis for Daniel Dennett's two-stage model of free will.


Bibliography


Poincaré's writings in English translation

Popular writings on the philosophy of science: *; reprinted in 1921; This book includes the English translations of Science and Hypothesis (1902), The Value of Science (1905), Science and Method (1908). * 1905. "," The Walter Scott Publishing Co. * 1906. "," Athenæum * 1913. "The New Mechanics," The Monist, Vol. XXIII. * 1913. "The Relativity of Space," The Monist, Vol. XXIII. * 1913. * 1956. ''Chance.'' In James R. Newman, ed., The World of Mathematics (4 Vols). * 1958. ''The Value of Science,'' New York: Dover. On
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
: * 1895. . The first systematic study of
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. On
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
: * 1890. * 1892–99. ''New Methods of Celestial Mechanics'', 3 vols. English trans., 1967. . * 1905. "The Capture Hypothesis of J. J. See," The Monist, Vol. XV. * 1905–10. ''Lessons of Celestial Mechanics''. On the philosophy of mathematics: * Ewald, William B., ed., 1996. ''From Kant to Hilbert: A Source Book in the Foundations of Mathematics'', 2 vols. Oxford Univ. Press. Contains the following works by Poincaré: ** 1894, "On the Nature of Mathematical Reasoning," 972–81. ** 1898, "On the Foundations of Geometry," 982–1011. ** 1900, "Intuition and Logic in Mathematics," 1012–20. ** 1905–06, "Mathematics and Logic, I–III," 1021–70. ** 1910, "On Transfinite Numbers," 1071–74. * 1905. "The Principles of Mathematical Physics," The Monist, Vol. XV. * 1910. "The Future of Mathematics," The Monist, Vol. XX. * 1910. "Mathematical Creation," The Monist, Vol. XX. Other: * 1904. ''Maxwell's Theory and Wireless Telegraphy,'' New York, McGraw Publishing Company. * 1905. "The New Logics," The Monist, Vol. XV. * 1905. "The Latest Efforts of the Logisticians," The Monist, Vol. XV. Exhaustive bibliography of English translations: * 1892–2017. .


See also


Concepts

* Hopf bifurcation, Poincaré–Andronov–Hopf bifurcation * Poincaré complex – an abstraction of the singular chain complex of a closed, orientable manifold * Poincaré duality * Poincaré disk model * Asymptotic expansion, Poincaré expansion * Gauge fixing#Multipolar gauge, Poincaré gauge *
Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
* Poincaré half-plane model * Homology sphere#Poincaré homology sphere, Poincaré homology sphere * Poincaré inequality * Closed and exact differential forms#Poincaré lemma, Poincaré lemma *
Poincaré map In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower-dimensiona ...
* Poincaré residue * Poincaré series (modular form) * Poincaré space * Poincaré metric * Poincaré plot * Betti number#Poincaré polynomial, Poincaré polynomial * Hilbert–Poincaré series, Poincaré series * Poincaré sphere (optics), Poincaré sphere * Einstein synchronisation, Poincaré–Einstein synchronisation * Poincaré–Lelong equation * Poincaré–Lindstedt method * Poincaré–Lindstedt perturbation theory * Poincaré–Steklov operator * Euler characteristic, Euler–Poincaré characteristic * Neumann–Poincaré operator * Reflecting Function


Theorems

Here is a list of theorems proved by Poincaré: * Poincaré's recurrence theorem: certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state. * Poincaré–Bendixson theorem: a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere. * Poincaré–Hopf theorem: a generalization of the hairy-ball theorem, which states that there is no smooth vector field on a sphere having no sources or sinks. * Poincaré–Lefschetz duality theorem: a version of Poincaré duality in geometric topology, applying to a manifold with boundary * Poincaré separation theorem: gives the upper and lower bounds of eigenvalues of a real symmetric matrix B'AB that can be considered as the orthogonal projection of a larger real symmetric matrix A onto a linear subspace spanned by the columns of B. * Poincaré–Birkhoff theorem: every area-preserving, orientation-preserving homeomorphism of an annulus that rotates the two boundaries in opposite directions has at least two fixed points. * Poincaré–Birkhoff–Witt theorem: an explicit description of the universal enveloping algebra of a Lie algebra. * Kelvin's circulation theorem#Poincaré–Bjerknes circulation theorem, Poincaré–Bjerknes circulation theorem: theorem about a conservation of quantity for the rotating frame. *
Poincaré conjecture In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured ...
(now a theorem): Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. * Poincaré–Miranda theorem: a generalization of the intermediate value theorem to ''n'' dimensions.


Other

*French epistemology *History of special relativity *List of things named after Henri Poincaré *Institut Henri Poincaré, Paris *Brouwer fixed-point theorem *Relativity priority dispute *Epistemic structural realism"Structural Realism"
entry by James Ladyman in the ''Stanford Encyclopedia of Philosophy''


References


Footnotes


Sources

* Eric Temple Bell, Bell, Eric Temple, 1986. ''Men of Mathematics'' (reissue edition). Touchstone Books. . * Belliver, André, 1956. ''Henri Poincaré ou la vocation souveraine''. Paris: Gallimard. *Peter L. Bernstein, Bernstein, Peter L, 1996. "Against the Gods: A Remarkable Story of Risk". (p. 199–200). John Wiley & Sons. * Carl Benjamin Boyer, Boyer, B. Carl, 1968. ''A History of Mathematics: Henri Poincaré'', John Wiley & Sons. * Ivor Grattan-Guinness, Grattan-Guinness, Ivor, 2000. ''The Search for Mathematical Roots 1870–1940.'' Princeton Uni. Press. * . Internet version published in Journal of the ACMS 2004. * Folina, Janet, 1992. ''Poincaré and the Philosophy of Mathematics.'' Macmillan, New York. * Jeremy Gray, Gray, Jeremy, 1986. ''Linear differential equations and group theory from Riemann to Poincaré'', Birkhauser * Gray, Jeremy, 2013. ''Henri Poincaré: A scientific biography''. Princeton University Press * * Kolak, Daniel, 2001. ''Lovers of Wisdom'', 2nd ed. Wadsworth. * Gargani, Julien, 2012. ''Poincaré, le hasard et l'étude des systèmes complexes'', L'Harmattan. * Murzi, 1998. "Henri Poincaré". * O'Connor, J. John, and Robertson, F. Edmund, 2002, "Jules Henri Poincaré". University of St. Andrews, Scotland. * Ivars Peterson, Peterson, Ivars, 1995. ''Newton's Clock: Chaos in the Solar System'' (reissue edition). W H Freeman & Co. . * Sageret, Jules, 1911. ''Henri Poincaré''. Paris: Mercure de France. * Toulouse, E.,1910. ''Henri Poincaré''.—(Source biography in French) at University of Michigan Historic Math Collection. * * F. Verhulst, Verhulst, Ferdinand, 2012 ''Henri Poincaré. Impatient Genius''. N.Y.: Springer. * ''Henri Poincaré, l'œuvre scientifique, l'œuvre philosophique'', by Vito Volterra, Jacques Hadamard, Paul Langevin and Pierre Boutroux, Felix Alcan, 1914. ** ''Henri Poincaré, l'œuvre mathématique'', by Vito Volterra. ** ''Henri Poincaré, le problème des trois corps'', by Jacques Hadamard. ** ''Henri Poincaré, le physicien'', by Paul Langevin. ** ''Henri Poincaré, l'œuvre philosophique'', by Pierre Boutroux. *


Further reading


Secondary sources to work on relativity

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *


Non-mainstream sources

* *


External links

* * *
Henri Poincaré's Bibliography
*Internet Encyclopedia of Philosophy:
Henri Poincaré
"—by Mauro Murzi. *Internet Encyclopedia of Philosophy:
Poincaré’s Philosophy of Mathematics
—by Janet Folina. *
Henri Poincaré on Information Philosopher
*
A timeline of Poincaré's life
University of Nantes (in French).
Henri Poincaré Papers
University of Nantes (in French).

*Collins, Graham P.,
Henri Poincaré, His Conjecture, Copacabana and Higher Dimensions
" ''Scientific American'', 9 June 2004. *BBC in Our Time,
Discussion of the Poincaré conjecture
" 2 November 2006, hosted by Melvynn Bragg.
Poincare Contemplates Copernicus
at MathPages
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. {{DEFAULTSORT:Poincare, Henri 1854 births 1912 deaths Henri Poincaré, 19th-century essayists 19th-century French male writers 19th-century French mathematicians 19th-century French non-fiction writers 19th-century French philosophers 20th-century essayists 20th-century French male writers 20th-century French mathematicians 20th-century French non-fiction writers 20th-century French philosophers Algebraic geometers Burials at Montparnasse Cemetery Chaos theorists Continental philosophers Corps des mines Corresponding members of the Saint Petersburg Academy of Sciences Deaths from embolism Determinists Dynamical systems theorists École Polytechnique alumni Fluid dynamicists Foreign associates of the National Academy of Sciences Foreign Members of the Royal Society French male essayists French male non-fiction writers French male writers French military personnel of the Franco-Prussian War French mining engineers French geometers Lecturers Mathematical analysts Members of the Académie Française Members of the Royal Netherlands Academy of Arts and Sciences Mines ParisTech alumni Officers of the French Academy of Sciences Scientists from Nancy, France Philosophers of logic Philosophers of mathematics Philosophers of psychology Philosophers of science Philosophy academics Philosophy writers Recipients of the Bruce Medal Recipients of the Gold Medal of the Royal Astronomical Society French relativity theorists Thermodynamicists Topologists University of Paris faculty Recipients of the Matteucci Medal