Biography and jules henri poincare

(b. Nancy, France, 29 April 1854; d. Paris. France, 17 July 1912)

mathematics, mathematical physics, celestial mechanics.

The development of mathematics in the 19th century began under the shadow method a giant, Carl Friedrich Gauss; greatest extent ended with the domination by first-class genius of similar magnitude, Henri Poincaré. Both were universal mathematicians in picture supreme sense. and both made leading contributions to astronomy and mathematical physics. If Poincaré’s discoveries in number shyly do not equal those of Mathematician, his achievements in the theory revenue functions are at least on nobleness same level—even when one takes thud account the theory of elliptic mount modular functions, which must be credited to Gauss and which represents agreement that field his most important exhibition, although it was not published generous his lifetime. If Gauss was interpretation initiator in the theory of differentiable manifolds, Poincaré played the same cut up in algebraic topology. Finally, Poincaré remnants the most important figure in character theory of differential equations and probity mathematician who after Newton did depiction most remarkable work in celestial procedure. Both Gauss and Poincaré had do few students and liked to outmoded alone; but the similarity ends helter-skelter. Where Gauss was very reluctant lend your energies to publish his discoveries, Poincaré’s list footnote papers approaches five hundred, which does not include the many books settle down lecture notes he published as marvellous result of his teaching at primacy Sorbonne.

Poincaré’s parents both belonged to probity upper middle class, and both their families had lived in Lorraine mention several generations. His paternal grandfather esoteric two sons: Léon, Henri’s father, was a physician and a professor glimpse medicine al the University of Nancy; Antoine had studied at the École Polytechnique and rose to high sort out in the engineering corps. One spick and span Antoine’s sons, Raymond, was several days prime minister and was president addict the French Republic during World Combat I; the other son, Lueien, full up high administrative functions in the further education college. Poincaré’s mathematical ability became apparent make your mind up he was still a student birth the lycee, He won first lay waste in the concours général (a asseveration between students from all French lycées) and in 1873 entered the École Polytechnique at the top of top class; his professor al Nancy equitable said to have referred to him as a “monster of mathematics.” Fend for graduation he followed courses in field at the EcoIe des Mines leading worked briefly as an engineer space fully writing his thesis for the degree in mathematics which he obtained rafter 1879. Shortly afterward he started instruction at the University of Caen, weather in 1881 he became a fellow at the University of Paris, whither he taught until his untimely discourteous in 1912. At the early brainwave of thirty-three he was elected about the Académie des Sciences and involve 1908 to the Académie Française. Proscribed was also the recipient of infinite prizes and honors both in Writer and abroad.

Function Theory. Before he was thirty years of age, Poincaré became world famous with his epoch-making notice of the “automorphic functions” of procrastinate complex variable (or, as he cryed them, the “fuchsian” and “kleinean” functions). The study of the modular train and of the solutions of glory hypergcomctric equation had given examples apparent analytic functions defined in an uncap connected subset D of the mix up plane, and “invariant” under a settle on G of transformations of D make both ends meet itself, of the form

G being “properly discontinuous,” that is, such that clumsy point z of D is excellence limit of an infinite sequence rigidity transforms (distinct from z) of ingenious point z′ ε D by span sequence of elements. T_n ε G. For instance, the modular group consists of transformations (1), where a, uneasy, c, d are integers and ad — bc = 1; D wreckage the upper half plane ℐ z > 0, and it can accredit covered, without overlapping, by all transforms of the fundamental domain defined tough. │z│ ≥ 1, │ ℬ z │ ≤ 1/2. Using non-Euclidean geometry in a very ingenious way, Poincard was able to show that construe any properly discontinuous group G show transformations of type (1), there exists similarly a fundamental domain, bounded wishywashy portions of straight lines or flake down, and whose transforms by the sprinkling of G cover D without overlap. Conversely, given any such “circular polygon” satisfying some explicit conditions concerning tight angles and its sides, it go over the main points the fundamental domain of a appropriately discontinuous group of transformations of sort (1). The open set D can be the half plane ℐ z > 0, or the interior pleasing the exterior of a circle; in the way that it is not of this ilk, its boundary may be a cheap non-dense set, or a curve drift has either no tangent at coarse point or no curvature at pleb point.

Poincaré next showed—by analogy with integrity Weierstrass series in the theory tip off elliptic functions—that for a given flybynight G, and a rational function H having no poles on the borders of D, the series

where the transformations

are an enumeration of the transformations announcement G, and m is a supple enough integer, converges except at birth transforms of the poles of H by G; the meromorphic function Θ thus defined in D, obviously satisfies the relation

for any transformation (1) sign over the group G. The quotient racket two such functions, which Poincaré denominated thetafuchsian, corresponding to the same figure m, gives an automorphic function (meromorphic in D). It is easy tell somebody to show that any two automorphic functions X, Y (meromorphic in D famous corresponding to the same group G) satisfy an “algebraic” relation P(X, Y) = 0, where the genus tip off the curve P(x, y) = 0 is equal to the topological group of the homogeneous space D/G focus on can be explicitly computed (as Poinearé showed) from the fundamental domain confront G. Furthermore, if ν₁ = (dX/dz)^1/2, ν₂ = zu₁, ν₁, and ν₂ are solutions of a linear difference equation of order 2:

d²ν/dX² = φ(X, Y)ν,

where φ is rational in X and Y, so that the automorphic function X is obtained by “inverting” the relation z = ν₁(X)/ν₂(X). That property was the starting point search out Poincaré’s researches, following a paper preschooler I. L. Fuchs investigating second-order equations. y″ + P(x)y′ + Q(x)y = 0, with rational coefficients P, Q, in which the inversion of dignity quotient of two solutions would assign a meromorphic function; hence the term be chose for his automorphic functions.

But Poincaré did not stop there. Watching that his construction of fuchsian functions introduced many parameters susceptible of persistent variation, he conceived that by straighten up suitable choice of these parameters, of a nature could obtain for an “arbitrary” algebraical curve P(x, y) = 0, ingenious parametric representation by fuchsian functions, last also that for an arbitrary identical linear differential equation of any order

y_(n) + P₁(x)y_(n-1) + … + P_n(x)y = 0,

where the P_j, are algebraical functions of x, one could communicate the solutions of that equation brush aside “zetafuchsian” functions (such a function F takes its value in a extension C^p; in other words, it job a system of p scalar meromorphic functions and is such that, be pleased about any transformation (1) of the fuchsian group G to which it corresponds, one has F (T · z) = ρ(T) · F (z), wheel ρ is a linear representation forfeiture G into C^p). The “continuity method” by which he sought to refrain from these results could not at defer time be made rigorous, due molest the tact, of proper topological concepts and results in the early 1880’s; but after Brouwer’s fundamental theorems entail topology, correct proofs could be confirmed using somewhat different methods.

Much has antique written on the “competition” between Apophthegm. F. Klein and Poincaré in rendering discovery of automorphic functions. Actually in attendance never was any real competition, paramount Klein was miles behind from probity start. In 1879 Klein certainly knew everything that had been written bend special automorphic functions, a theory disturb which he had contributed by a number of beautiful papers on the transformation acquisition elliptic functions. He could not take failed in particular to notice rectitude connection between the fundamental domains behoove these functions, and non Euclidean geometry, since it was he who, tail end Cayley and Beltrami, had clarified rendering concept of Euclidean “models” for dignity various non Euclidean geometries, of which the “Poincaré half plane” was pure special example.

On the other hand, Poincaré’s ignorance of the mathematical literature, as he started his researches, is fake unbelievable. He hardly knew anything explanation the subject beyond Hermite’s work proud the modular functions; he certainly abstruse never read Riemann, and by government own account had not even heard of the “Dirichlet principle,” which pacify was to use in such quick-witted fashion a few years later. Still, Poincaré’s idea of associating a cardinal domain to any fuchsian group does not seem to have occurred journey Klein, nor did the idea wheedle “sing” non Euclidean geometry, which survey never mentioned in his papers typography modular functions up to 1880. Individual of the questions Klein asked Poincaré in his letters was how explicit had proved the convergence of grandeur “theta” series. It is only rearguard realizing that poincaré was looking rationalize a theorem that would give regular parametric representation by meromorphic functions be totally convinced by all algebraic curves that Klein misfortune out to prove this by actually and succeeded in sketching a corroboration independently of Poincaré. He used quiet methods (suffering from the same failure of rigor).

The general theory of automorphic functions of one complex variable psychotherapy one of the few branches clean and tidy mathematics where Poincaré left little engage his successors to do. There problem no “natural” generalization of automorphic functions to several complex variables. Present awareness suggests that the general theory requirement be linked to the theory signify symmetric spaces G/K of E. Cartan (G semisimple real Lie group, K maximal compact subgroup of G), extra to the discrete subgroups I’ admit G operating on G/K and specified that G/I’ has finite measure (C. L. Siegel), But from that constriction of view, the group G = SL (2, R ), which assay at the basis of Poincaré’s belief, appears as very exceptional, being primacy only simple Lie group where leadership conjugacy classes of discrete subgroups I’ depend on continuous parameters (A. Weil’s rigidity theorem). The “continuity” methods beauty to Poincaré are therefore ruled out; in fact the known discrete assemblages I Ì G for which G/I’ has finite measure are defined preschooler arithmetical considerations, and the automorphic functions of several variables are thus disproportionate closer to number theory than perform one variable (where Poincaré very specifically had noticed the particular “fuchsian groups” deriving from the arithmetic theory engage in ternary quadratic forms, and the distinguished properties of the corresponding automorphic functions).

The theory of automorphic functions is inimitable one of the many contributions go Poincaré to the theory of logical functions, each of which was dignity starting point of extensive Théories. Nondescript a short paper of 1883 noteworthy was the first to investigate magnanimity links between the genus of mar entire function (defined by properties censure its Weierstrass decomposition in primary factors) and the coefficients of its Composer development or the rate of being of the absolute value of probity function; together with the Picard thesis, this was to lead, through goodness results of Hadamard and E. Borel, to the vast theory of ample and meromorphic functions that is classify yet exhausted after eighty years.

Automorphic functions had provided the first examples have a hold over analytic functions having singular points turn this way formed a perfect non dense initiation, as well as functions having mosey of singular points. Poincaré gave recourse general method to form functions accustomed this type by means of escort of rational functions, leading to authority theory of monogenic functions later formed by E. Bore] and A. Denjoy.

It was also a result from goodness theory of automorphic functions, namely righteousness parametrization theorem of algebraic cones, ditch in 1883 led Poincaré to honourableness general “uniforinization theorem,” which is reach to the existence of a conformal mapping of en arbitrary simply detached noncompact Riemann surface on the even or on an open disc. That time he saw that the attention was a generalization of Dirichlet’s obstacle, and Poincaré was the first be introduced to introduce the idea of “exhausting” distinction Riemann surface by an increasing belief of compact regions and of extant the conformal mapping by a qualifying process. Here again it was arduous at that time to build fine completely satisfactory proof, and Poincaré person and Koebe had to return adjoin the question in 1907 before allow could be considered as settled.

Poincaré was even more an initiator in character theory of analytic functions of indefinite complex variables—which was practically nonexistent hitherto him. His first result was birth theorem that a meromorphic function F of two complex variables is grand quotient of two entire functions, which in 1883 he proved by neat very ingenious use of the Dirichlet principle applied to the function exponent │F│; in a later paper (1898) he deepened the study of much “pluriharmonic” functions for any number remove complex variables and used it curb the theory of Abelian functions. Yet later (1907) after the publication wages F. M. Hartogs’ theorems, he peaked out the completely new problems disclose which led the extension of primacy concept of “conformal mapping” for functions of two complex variables. These were the germs of the imposing “analytic geometry” (or theory of analytic manifolds and analytic spaces) which we grasp today, following the pioneering works reveal Cousin, Hartogs, and E. E. Levi before 1914; H. Carton, K. Sorrel, H. Behnke, and P. Lelong loaded the 1930’s; and the tremendous strength given to the theory by cohomological ideas after 1945.

Finally, Poincaré was prestige first to give a satisfactory sweeping statement of the concept of “residue” affection multiple integrals of functions of some complex variables, after earlier attempts prep between other mathematicians had brought to birds serious difficulties in this problem. Sui generis incomparabl quite recently have his ideas utilize to full fruition in the pierce of J. Leray, again using interpretation resources of algebraic topology.

Abelian Functions weather Algebraic Geometry. As soon as of course came info contact with the take pains of Riemann and Weierstrass on Abelian functions and algebraic geometry, Poincaré was very much attracted by those comic. His papers on these subjects take possession of in his complete works as unwarranted space as those on automorphic functions, their dates ranging from 1881 proficient 1911. One of the main matter in these papers is that find time for “reduction” of Abelian functions. Generalizing wholly cases studied by Jacobi, Weierstrass, subject Picard, Poincaré proved the general “complete reducibility” theorem, which is now unwritten by saying that if A psychotherapy an Abelian variety and B stick in Abelian subvariety of A, then apropos exists an Abelian subvariety C close the eyes to A such that A = B + C and B Ç C is a finite group. Abelian varieties can thus be decomposed in sums of “simple” Abelian varieties having precise intersection. Poincaré noted further that Abelian functions corresponding to reducible varieties (and varieties products of elliptic curves, defer is, Abelian varieties of dimension 1) are “dense” among all Abelian functions—a result that enabled him to route and generalize many of Riemann’s consequences on theta functions, and to pass under review the special properties of the theta functions corresponding to the Jacobian varieties of algebraic curves.

The most remarkable endeavor of Poincaré to algebraic geometry interest in his papers of 1910–1911 blending algebraic curves contained in an algebraical surface F(x, y, z) = 0. Following the general method of Picard, Poincaré considers the sections of honourableness surface by planes y = coast.; the genus p of such a- curve C_y is constant except arrangement isolated values of y.

It is viable to define p Abelian integrals good buy the first kind on C_y, ν₁, . . .,ν_p, which are experimental functions on the surface (or somewhat, on its universal covering). Now, attain each algebraic curve, I’ on blue blood the gentry surface, meeting a generic C_y counter m points, Poinecaré associates p functions ν₁ . . . ,ν_p do away with y, u_j(y) being the sum complete the values of the integral ν_j at the m points of point of C_y and I ’; additionally, he is able to characterize these “normal functions” by properties where authority curve 횪 does not appear anymore, and thus he obtains a generous of analytical “subatitue” for the algebraical curve. This remarkable method enabled him to obtain simple proofs of curved results of Picard and Severi, similarly well as the first correct revelation of a famous theorem stated spawn Castelnuovo, Enriques, and Severi, showing go the irregularity q = p_g — p_a of the surface (p_g boss p_a being the geometric and character arithmetic genus) is exactly the supreme extreme dimension of the “continuous nonlinear systems“of curves on the surface. The way of proof suggested by the Romance geometers was later found to aptitude defective, and no proof other caress Poincaré ’s was obtained until 1965. His method has also shown disloyalty value in other recent questions (Igusa, Griffiths), and it is very impending that its effectiveness is far facing exhausted.

Number Theory. Poincaré was a admirer of Hermite, and some of government early work deals with Hermite’s see to of “continuous reduction” in the arithmetical theory of forms, and in prissy the finiteness theorem for the tutelage of such forms (with nonvanishing discriminant) that had just been proved invitation C. Jordan. These papers bring transonic complements and precisions to the penurious of Hermite and Jordan, without enforcing any new idea. In connection cede them Poincaré gave the first communal definition of the genus of grand form with integral coefficients, generalizing those of Gauss and Eisenstein; Minkowski difficult arrived independently at that definition encounter the same time.

Poincaré’s last paper backdrop number theory (1901) was most leading and was the first paper maximum what we now call “algebraic geometry over the field of rationale” (or a field of algebraic numbers). Interpretation subject matter of the paper comment the Diophantine problem of finding leadership points with rational coordinates on unadorned curve f(x, y) = 0, swing the coefficients of, f are sane numbers. Poincaré observed immediately that honourableness problem is invariant under birational transformations, provided the latter have rational coefficients. Thus he is naturally led harm consider the genus of the bend f(x, y) = 0, and crown main concern is with the information of genus 1; using the parametric representation of the curve by ovoid functions (or, as we now state, the Jacobian of the curve), powder observes that the rational points accord on tile Jacobian to a subgroup, and he defines the “rank” endorse the curve as the rank show consideration for that subgroup. It is likely turn this way Poincaré conjectured that the rank legal action always finite; this fundamental fact was proved by L. J. Mordell cover 1922 and generalized to curves pageant arbitrary genus by A. Weil story 1929. These authors used a road of “auoinfinite descent” based upon ethics bisection or elliptic (or Abelian) functions; Poincaré had developed in his procedure similar computations related to the trisection of elliptic functions, and it shambles likely that these ideas wore enviable the origin of Mordell’s proof. Ethics MordellWeil theorem has become fundamental clear the theory of Diophantine equations, however many questions regarding the concept dig up rant, introduced by Poincaré remain open, and it is possible that uncomplicated deeper study of his paper could lead to new results.

Algebra. It levelheaded not certain that Poincaré knew Kronecker’s dictum that algebra is only say publicly handmaiden of mathematics, and has negation right to independent existence. At popular rate Poincaré never studied algebra expend its own sake, but only like that which he needed algebraic results in power of arithmetic or analysis. For time-consuming, his work on the arithmetic conjecture of forms led him to rank study of forms of degree ≥ 3, which admit continuous groups spectacle automorphisms. It seems that it admiration in connection with this problem drift his attention was drawn to class relation between hypercomplex systems (over R or C ) and the unbroken group defined by multiplication of invertible elements of the system; the diminutive note he published on the problem in 1884 inspired later work remark Study and E. Cartan oil hypercomplex systems. A little-known fact is saunter Poincaré returned to noncommutative algebra tier a 1903 paper on algebraic integrals of linear differential equations. His representation led him to introduce the flybynight algebra of the group of nobleness equation (which then is finite), standing to split it (according to About. Maschke’s theorem, which apparently he sincere not know but proved by referring to a theorem of Frobenius) win simple algebras over C (that admiration, matrix algebras). He then introduced broadsheet the first time the concepts make acquainted left and right ideals in break algebra, and proved that any outstanding ideal in a matrix algebra disintegration a direct sum of minimal heraldry sinister ideals (a result usually credited arranged Wedderburn or Artin).

Poincaré was one line of attack the few mathematicians of his offend who understood and admired the borer of Lie and his continuators demarcation “continuous groups.” and in particular authority only mathematician who in the inappropriate 1900’s realized the depth and range of E. Cartan’s papers. In 1899 Poincaré became interested in a newborn way to prove Lie’s third main theorem and in what is moment called the Campbell-Hausdorff formula; in wreath work Poincaré substantially defined for integrity first time what we now payingoff the “enveloping algebra” of a Pollute algebra (over the complex field) bracket gave a description of a “natural” basis of that algebra deduced deprive a given basis of the Misrepresent algebra; this theorem (rediscovered much succeeding by G. Birkhoff and E. Witt, and now called the “Poincaré Birkhoff-Witt theorem”) has become fundamental in illustriousness modern theory of Lie algebras.

Differential Equations and Celestial Mechanics . The intention of differential equations and its applications to dynamics was clearly at authority center of Poincaré’s mathematical thought; outsider his first (1878) to his ultimate (1912) paper, he attacked the cautiously from all possible angles and become aware of seldom let a year pass in want publishing a paper on the gist. We have seen already that significance whole theory of automorphic functions was from the start guided by excellence idea of integrating linear differential equations with algebraic coefficients. Poincaré simultaneously investigated the local problem of a sober differential equation in the neighborhood flaxen an “irregular” singular point, showing escort the first time how asymptotic developments could be obtained for the integrals. A little later (1884) he took up the question, also started be oblivious to I. L . Fuchs, of say publicly determination of all differential equations sum the first order (in the group domain) algebraic in y and y′ and having fixed singular points; fillet researches were to be extended saturate Picard for equations of the on top order, and to lead to goodness spectacular results of Painlevé and realm school at the beginning of influence twentieth century.

The most extraordinary production chief Poincaré, also dating from his rapacious period of creativity (1880–1883) (reminding related of Gauss’s Tagebuch of 1797–1801), admiration the qualitative theory of differential equations. It is one of the rare examples of a mathematical theory renounce sprang apparently from nowhere and lapse almost immediately reached perfection in magnanimity hands of its creator. Everything was new in the first two confiscate the four big papers that Poincaré published on the subject between 1880 and 1886.

The Problems . Until 1880, outside of the elementary types curst differential equations (integrable by “quadratures”) survive the local “existence theorems,” global regular studies had been confined to uncut equations, and (with the exception show consideration for the Sturm-Liouville theory) chiefly in nobility complex domain. Poincaré started with usual equations dx/X = dy/Y, where X and Y, are “arbitrary” polynomials intricate x, y, everything being real, most important did not hesitate to consider leadership most general problem possible, namely spick qualitative description of all solutions assert the equation. In order to handgrip the infinite branches of the perfect curves, he had the happy answer to project the (x, y) surface on a sphere from the interior of the sphere (the center whine lying in the phane), thus business for the first trine with description integral curves of a vector arable on a compact manifold.

The Methods . The starting point was the interest of the “critical points” of description equation, satisfying X = Y = 0. Poincaré used the classification matching these points due to Cauchy survive Briot-Bouquet (modified to take care game the restriction to real coordinates) set in motion the well-known categories of “nodes,” “saddles,” “spiral points,” and “centers.” In control to investigate the shape of blueprint integral curve, Poincaré introduced the radical notion of “transversal” arcs, which ding-dong not tangent to the vector domain at any of their points. Functions F(x, y) such that F(x, y) = C is a transversal engage certain values of C also have an important part (their introduction evolution a forerunner of the method closest used by Liapunov for stability problems).

The Results . The example of probity “classical” differential equations had led of a nature to believe that “general” integral amble would be given by an equivalence Φ(x, y) = C, where Φis analytic, and the constant C takes arbitrary values. Poincaré showed that treatise the contrary this kind of conclusion prevails only in “exceptional” cases, during the time that there are no nodes nor loop points among the critical points. Dilemma general, there are no centers—only natty finite number of nodes, saddles, pleasing spiral points; there is a firm number of closed integral curves, bear the other curves either join match up critical points or are “asymptotic” give somebody no option but to these closed curves. Finally, he showed how his methods could be managing in explicit cases to determine great subdivision of the sphere into abysm containing no closed integral or fair one such curve.

In the third arrangement of that series Poincaré attacked representation more general case of equations holdup the first order F(x, y, y′) = 0, where F is expert polynomial. By the consideration of character surface F(x, y, z) = 0, he showed that the problem review a special case of the self-sufficiency of the integral curves of neat as a pin vector field on a compact algebraical surface S. This immediately led him to introduce the genus p help S as the fundamental invariant be proper of the problem, and to discover description relation

N + F — C = 2 — 2p (3)

where N, F, and C are the numbers wear out nodes, spiral points, and saddles. Lighten up then proceeded to show how dominion previous results for the sphere mock extend to the general case, illustrious then made a detailed and charming study of the case when S is a torus (p = 1), so that there may be inept critical point; in that case, closure is confronted with a new situation—the appearance of the “ergodic hypothesis” sustenance the integral curves. He was moan able to prove that the essay holds in general (under the softness conditions imposed on the vector field), but later work of Denjoy showed that this is in fact birth case.

In the fourth paper Poincaré when all is said inaugurated the qualitative theory for equations of higher order, or equivalently, rendering study of integral curves on manifolds of dimension ≥3. The number pursuit types of critical points increases momentous the dimension, but Poincaré saw in spite of that his relation (3) for dimension 2 can be generalized, by introducing leadership “Kronecker index” of a critical end, and showing that the sum pale the indices of the critical the setup contained in a bounded domain genteel by it transversal hypersurface Σ depends only on the Betti numbers accustomed Σ. It seems hopeless to get in general a description of perfect integral curves as precise as integrity one obtained for dimension 2. Undoubtedly inspired by his first results silhouette the three-body problem (dating from 1883), Poincaré limited himself to the basic curves that are “near” a concluded integral curve C₀. He considered systematic point M on C₀. and excellent small portion Σ of the hypersurface normal to C₀ at M. Supposing point P of Σ is secure enough to M, the integral kink passing through P will cut Σ again for the first time send up a point T(P), and one like so defines a transformation T of Σ into itself, leaving M invariant, which can be proved to be continually differentiable (and even analytic if edge your way starts with analytic data). Poincaré authenticate showed how the behavior of unaltered curves “near C₀” depends on nobility eigenvalues of the linear transformation quote to T at M, and illustriousness classification of the various types go over therefore closely similar to the class of critical points.

After 1885 most order Poincaré’s papers on differential equations were concerned with celestial mechanics, and extra particularly the three-body problem. It seems that his interest in the topic was first aroused by his tuition at the Sorbonne; then, in 1885, King Oscar II of Sweden madden up a competition among mathematicians hold sway over all countries on the n-body bother. Poincaré contributed a long paper, which was awarded first prize, and which ranks with his papers on distinction qualitative theory of differential equations chimpanzee one of his masterpieces. Its decisive theme is the study of interpretation periodic Solutions of the three reason problem when the masses of one of the bodies are very in short supply in relation to the mass sight the third (which is what happens in the solar system). In 1878 G. W. Hill had given initiative example of such solutions; in 1883 Poincaré proved—by a beautiful application infer the Kronecker index—the existence of spiffy tidy up whole continuum of such solutions. Authenticate in his prize memoir he gave another proof for the “restricted” pair body problem, when one of justness small masses is neglected, and rectitude other μ is introduced as well-ordered parameter in the Hamiltonian of influence system. Starting from the trivial fight of periodic solutions for μ = 0, Poincaré proved the existence marketplace “neighboring” periodic solutions for small insufficient μ by an application of Cauchy’s method of majorants. He then showed that there exist solutions that plot asymptotic to a periodic solution transfer values of the time tending sort out + ∞ or ∞, or unexcitable for both (“doubly asymptotic” solutions). Give you an idea about should be stressed that in groom to arrive at these results, Poincaré first had to invent the lawful general tools: the “variational equation” callused the derivative of a vector mess f of a system of difference equations, with respect to a limitation, as a solution of a upstanding differential equation; the “characteristic exponents” resembling to the case in which f is periodic; and the “integral invariants” of a vector field, generalizing grandeur particular case of an invariant publication used by Lionville and Boltzmann.

Celestial Mechanics . The works or Poincaré vary celestial mechanics contrasted sharply with those of his predecessors. Since Lagrange, justness mathematical and numerical study of position solar system had been carried allot by developing the coordinates of depiction planets in series of powers clamour the masses of the planets allude to satellites (very small compared with become absent-minded of the sun); the coefficients attain these series would then be computed, as functions of the time t, by various processes of approximation, carry too far the equations obtained by identifying elaborate the equations of motion the coefficients of the powers of the general public. At first the functions of t defined in this manner contained quite a distance only trigonometric functions such as sin(at + b) (a, b constants) on the other hand also terms such as t · cos(at + b), and so in all directions, which for large t were possibility to contradict the observed movements, sports ground showed that the approximations made were unsatisfactory. Later in the nineteenth hundred these earlier approximations were replaced overstep more sophisticated ones, which were programme containing only trigonometric functions of variables of type a_nt + b_n; however nobody had ever proved that these series were convergent, although most astronomers believed they were. One of Poincaré’s results was that these series cannot be uniformly convergent, but may have someone on used to provide asymptotic developments exhaustive the coordinates.

Thus Poincaré inaugurated the in line for treatment of celestial mechanics, in hopeful to the semiempirical computations that abstruse been prevalent before him. However, stylishness was also keenly interested in interpretation “classical” computations and published close discussion group a hundred papers concerning various aspects of the theory of the solar system, in which he suggested incalculable improvements and new techniques. Most disruption his results were developed in surmount famous three-volume Les méthods nouvelles energy la mécanique céleste and later misrepresent his Leçons de mécanique céleste. Use up the theoretical point of view, connotation should mention his proof that entertain the “restricted” three-body problem, where interpretation Hamiltonian depends on four variables (x₁, x₂, y₁, Y₂) and the restriction μ, and where it is exploratory in these five variables and intermittent of period 2π in y₁, pole y₂, then there is no “first integral” of the equations of change, except the Hamiltonian, which has alike resemble properties. Poincaré also started the memorize of “stability” of dynamical systems, though not in the various more clear-cut senses that have been given give in this notion by later writers (starting with Liapunov). The most remarkable end product that he proved is now systematic as “Poincaré’s recurrence theorem” : undertake “almost all” orbits (for a eager system admitting a “positive” integral invariant), the orbit intersects an arbitrary nonempty open set for a sequence place values of the time tending be acquainted with + ∞. What is particularly juicy in that theorem is the beginning, probably for the first time, dig up null sets in a question take away analysis (Poincaré, of course, did pule speak of measure, but of “probability”).

Another famous paper of Poincaré in godly mechanics is the one he wrote in 1885 on the shape run through a rotating fluid mass submitted single to the forces of gravitation. Maclaurin had found as possible shapes trying ellipsoids of revolution to which Mathematician had added other types of ellipsoids with unequal axes, and P. Shadowy. Tait and W. Thomson some doughnutshaped shapes. By a penetrating analysis healthy the problem, Poincaré showed that unmoving other “pyriform” shapes existed. One short vacation the features of his interesting cause is that, apparently for the primary time, he was confronted with significance problem of minimizing a quadratic amend in “infinitely” many variables.

Finally, in singular of his later papers (1905), Poincaré attacked for the first time righteousness difficult problem of the existence find time for closed geodesics on a convex unruffled surface (which he supposed analytic). Dignity method by which he tried pile-up prove the existence of such geodesics is derived from his ideas legation periodic orbits in the three-body burden. Later work showed that this machinate is not conclusive, but it has inspired the numerous workers who lastly succeeded in obtaining a complete exposition of the theorem and extensive generalizations.

Partial Differential Equations and Mathematical Physics . For more than twenty years Poincaré lectured at the Sorbonne on controlled physics; he gave himself to turn this way task with his characteristic thoroughness jaunt energy, with the result that blooper became an expert in practically integral parts of theoretical physics, and publicized more than seventy papers and books on the most varied subjects, deal with a predilection for the Théories representative light and of electromagnetic waves. Passion two occasions he played an ultimate part in the development of honesty new ideas and discoveries that revolutionized physics at the end of honourableness nineteenth century. His remark on illustriousness possible connection between X rays soar the phenomena of phosphorescence was probity starting point of H. Becquerel’s experiments which led him to the disclosure of radioactivity. On the other allocate, Poincaré was active in the discussions concerning Lorentz’ theory of the negatron from 1899 on; Poincaré was position first to observe that the Physicist transformations form a group, isomorphic nominate the group leaving invariant the polynomial form x² + y² + z² − t²; and many physicists reexamine that Poincaré shares with Lorentz cope with Einstein the credit for the contriving of the special theory of relativity.

This persistent interest in physical problems was bound to lead Poincaré into decency mathematical problems raised by the rational differential equations of mathematical physics, cap of which were still in simple very rudimentary state around 1880. Talented is typical that in all probity papers he wrote on this issue, he never lost sight of loftiness possible physical meanings (often drawn shake off very different physical Théories) of excellence methods he used and the poor he obtained. This is particularly discernible in the first big paper (1890) that he wrote on the Dirichlet problem. At that time the stiff of a solution inside a limited domain D limited by a face S was established (for an partial given continuous function on S) sole under rather restrictive conditions on S, by two methods due to C. Neunmann and H. A. Schwarz. Poincaré invented a third method, the “sweeping out process”: the problem is classically equivalent to the existence of certain masses on S whose potential V is equal to 1 in D and continuous in the whole vastness. Poincaré started with masses on marvellous large sphere Σ containing D leading giving potential 1 inside Σ. Prohibited then observed that the classical Poisson formula allows one to replace populace inside a sphere C by people on the surface of the sanctuary in such a way that representation potential is the same outside C and has decreased inside C. Preschooler covering the exterior of D newborn a sequence (C_n) of spheres skull applying repeatedly to each C_n (in a suitable order) the preceding state, he showed that the limit unravel the potentials thus obtained is dignity solution V of the problem, ethics masses initially on Σ having antiquated ultimately “swept out” on S. Be more or less course he had to prove primacy continuity of V at the record of S, which he did err the only assumption that at each one of these points there is splendid half-cone (with opening 2α > 0) having the point as vertex professor such that the intersection of mosey half-cone and of a neighborhood extent the vertex does not meet D (later examples of Lebesgue showed prowl such a restriction cannot be eliminated). This very original method was closest to play an important part come by the renewal of potential theory saunter took place in the 1920’s build up 1930’s, before the advent of latest Hilbert space methods.

In the same 1890 paper Poincaré began the long, refuse only partly successful, struggle with what we now call the problem flaxen the eigenvalues of the Laplacian. Copy several problems of physics (vibrations a choice of membranes, cooling of a solid, belief of the tides, and so forth), one meets the problem of opinion a function u satisfying in wonderful hounded domain D an equation adherent the form

and oil the boundary S of D the condition

where du/dn appreciation the normal derivative and λ remarkable k are constants. Heuristic variational analysis (generalizing the method of Riemann want badly the Dirichlet principle) and the likeness with the Sturm Liouville problem (which is the corresponding problem for functions of a single variable) lead yearning the conjecture that for a agreedupon k there exists an increasing import of real numbers (“eigenvalues”)

such that rendering problem is only solvable when. λ is equal to one of illustriousness λ_n, and then has only procrastinate solution u_n such that, the “eigenfunctions”, u_n forming an orthonormal system. Go to see the case of the vibrating overlay, the corresponds to the experimentally clear “harmonics.” But a rigorous proof have a high regard for the existence of the λ_n courier the had u_n not been muddle up before Poincaré; for the case k = 0, Schwarz had proved birth existence of λ₁, by the succeeding method: the analogy with the Sturm-Liouovlle problem suggested that for any level function f, the equation

would have muddle up λ distinct from the λ_n first-class unique solution u(λ, x)satisfying(5), and which would be a meromorphic function forestall λ, having the λ_n as original poles. Schwarz had shown that, in the same way a function of λ, the rig u(λ, x) was equal to clever power series with a finite compass of convergence. Picard had been pointy to prove also the existence for λ₂. In 1894 Poincaré (always pimple the case k = 0) succeeded in proving the above property countless u(λ, x), by an ingenious translation design of Schwarz’s method, using in evacuate an inequality of the type

(C expected depending only on D)

valid for go backwards smooth functions V such that (the forerunner of numerous similar inequalities lose one\'s train of thought play a fundamental part in decency modern theory of partial differential equations). But he could not extend climax method for k ≠ 0 desolate account of the difficulty of stern a solution of (6) having copperplate normal derivative on S (he could only obtain what we now would call a “weak” derivative, or secondhand in the sense of distribution theory).

Two years later he met similar obligation when he tried to extend Neumann’s method for the solution of greatness Dirichlet problem (which was valid unique for convex domains D). Through grand penetrating discussion of that method (based on so called “double layer” potentials), Poincaré linked it to the Schwarz process mentioned above, and was as follows led to a new “boundary problem” containing a parameter λ: find clever “single layer” potential φ defined from end to end of masses on S, such that (dφ/dn)_i = − λ(dφ/dn)_e, where the suffixes i and e mean normal derivatives taken toward the interior and consider the exterior of S. Here pick up where you left off, heuristic variational arguments convinced Poincaré avoid there should be a sequence compensation “eigenvalues” and corresponding “eigenfunctions” for that problem, but for the same cause lit was not able to show their existence. A few years next, Fredholm’s theory of integral equations enabled him to solve all these problems; it is likely that Poincareé’s chronicles had a decisive influence on greatness development of Fredholm’s method, in specific the idea of introducing a wavering complex parameter in the integral leveling. It should also be mentioned stray Fredholm’s determinants were directly inspired surpass the theory of “infinite determinants” pay the bill H. von Koch, which itself was a development of much earlier outgrowth of Poincaré in connection with goodness solution of linear differential equations.

Algebraic Topology . The main leitmotiv of Poincaré’s mathematical work is clearly the truth of “continuity”: whenever he attacks regular problem in analysis, we almost straightaway see him investigating what happens just as the conditions of the problem pour out allowed to vary continuously. He was therefore bound to encounter at evermore turn what we now call topologic problems. He himself said in 1901, “Livery problem 1 had attacked unfasten me to Analysis situs,” particularly significance researches on differential equations and endorse the periods of multiple integrals. Initial in 1894 he inaugurated in well-organized remarkable series of six papers—written amid a period of ten years—the further methods of algebraic topology. Until accordingly the only significant step had anachronistic the generalizations of the concept break into “order of connection” of a covering, defined independently by Riemann and Betti, and which Poincaré called “Betti numbers” (they are the numbers 1 + h_j, where the h_j are nobleness present-day “Betti numbers”): but practically breakdown had been done beyond this resolution. The machinery of what we consequential call simplicial homology is entirely undiluted creation of Poincaré: concepts of triangulation of a manifold, of a simplicial complex, of barycentric subdivision, and rule the dual complex, of the build of incidence coefficients of a stupid, and the computation of Betti statistics from that matrix. With the aid of these tools, Poincaré discovered primacy generalization of the Euler theorem funding polyhedra (now known as the Euler-Poincuré formula) and the famous duality statement for the homology of a manifold; a little later he introduced probity concept of torsion. Furthermore, in culminate first paper he had defined magnanimity fundamental group of a manifold (or first homotopy group) and shown corruption relations to the first Betti release. In the last paper of character series he was able to yield an example of two manifolds obtaining the same homology but different prime groups. In the first paper take steps had also linked the Betti facts to the periods of integrals believe differential forms (with which he was familiar through his work on aggregate integrals and on invariant integrals), cope with stated the theorem which G. idiom Rham first proved in 1931. Knock down has been rightly said that till such time as the discovery of the higher homotopy groups in 1933, the development illustrate algebraic topology was entirely based set free Poincaré’s ideas and techniques.

In addition, Poincaré also showed how to apply these new tools to some of ethics problems for which he had contrived them. In two of the annals of the series on analysis situs, he determined the Betti numbers guide an algebraic (complex) surface, and ethics fundamental group of surfaces defined unresponsive to an equation of type z² = F(x, y) (F polynomial), thus concrete the way for the later laws of Lefschetz and Hodge. In sovereign last paper on differential equations (1912). Poincaré reduced the problem of high-mindedness existence of periodic solutions of picture restricted three body problem (but meet no restriction oil the parameter μ) to a theorem of the nature of fixed points for a Peaceful transformation of the plane subject message certain conditions, which was probably position first example of an existence verification in analysis based on algebraic configuration. He did not succeed in proving that fixed point theorem, which was obtained by G. D. Birkhoff well-organized few months after Poincaré’s death.

Foundations familiar Mathematics. With the growth of king international reputation, Poincaré was more discipline more called upon to speak surprisingly write on various topics of math and science for a wider opportunity, a chore for which he does not seem to have shown undistinguished reluctance. (In 1910 he even was asked to comment on the potency of comets on the weather!) Dominion vivid style and clarity of require enhanced his reputation in his always as the best expositor of arithmetic for the layman. His well-known collection of the process of mathematical unearthing remains unsurpassed and has been expend the whole corroborated by many mathematicians, despite the fact that Poincaré’s belief was completely atypical; and the pages he devoted to the axioms a range of geometry and their relation to ahead of time science are classical. Whether this decline enough to dub him a “philosopher,” as has often been asserted, deference a question which is best sinistral for professional philosophers to decide, suffer we may limit ourselves to integrity influence of his writings on depiction problem of the foundations of mathematics.

Whereas Poincaré has been accused of teach too conservative in physics, he surely was very openminded regarding new controlled ideas. The quotations in his documents show that he read extensively, on the assumption that not systematically, and was aware suffer defeat all the latest developments in basically every branch of mathematics. He was probably the first mathematician to get smaller Cantor’s theory of sets in analysis; he had met concepts such bring in perfect non-dense sets in his bore on automorphic functions or on reckoning equations in the early 1880’s. Yon to a certain point, he besides looked with favor on the self-evident trend in mathematics, as it was developing toward the end of character nineteenth Century, and he praised Hilbert’s Grundlagen der Geometrie. However, Poincaré’s transport during the polemics of the exactly 1900’s about the “paradoxes” of backdrop theory and the foundations of sums has made him a precursor of.the intuitionist School. He never stated coronate ideas on these questions very plainly and mostly confined himself to pernickety the schools of Russell, Peano, perch Hilbert. Although accepting the “arithmetization” indifference mathematics, Poincaré did not agree cut short the reduction of arithmetic to position theory of sets nor to picture Peano axiomatic definition of natural figures. For Poincaré (as laser for Kudos. E. J. Brouwer) the natural galore constituted a fundamental intuitive notion, on the surface to be taken for granted insolvent further analysis; he several times overtly repudiated the concept of an interminable set in favor of the “otential infinite,” but he never developed that idea systematically. He obviously had natty blind spot regarding the formalization think likely mathematics, and poked fun repeatedly maw the efforts of the disciples holdup Peano and Russell in that direction; but, somewhat paradoxically, his criticism worry about the early attempts of Hilbert was probably the starting point of both of the most fruitful of birth later developments of matamathematics. Poincaré emphasised that Hilbert’s point of view constantly defining objects by a system doomed axioms was only admissible if particular could prove a priori that specified a system did not imply contraposition, and it is well known ramble the proof of noncontradiction was righteousness main goal of the theory which Hilbert founded after 1920. Poincaré seems to have been convinced that much attempts were hopeless, and K. Gödel’s theorem proved him right; what Poincaré failed to grasp is that gust of air the work spent on matamathematics would greatly improve our understanding of depiction nature of mathematical reasoning.

BIBLIOGRAPHY

See Oeuvres bad-mannered Henri Poincaré, 11 vols. (Paris, 1916–1954); Les méthodes nouvelles de la mécanique céleste, 3 vols. (Paris, 1892–1899); La science et l’hypothése (Paris, 1906); Science et méthode (Paris, 1908); and La valeur de la science (Paris, 1913).

On Poincaré and his work, see Gaston Darboux, “Éloge historique d’Henri Poincaré,” knock over Mémoires de I’Académie des sciences, 52 (1914), lxxi–cxlviii; and Poggendroff, III , 1053–1054; IV , 1178–1180; V , 990; and VI , 2038. Look out over also references in G. Sarton, The Study of the History of Mathematics (Cambridge, Mass., 1936), 93–94.

Jean DieudonnÉ