c Pleiades Publishing, Ltd., 2014. ISSN 1560-3547, Regular and Chaotic Dynamics, 2014, Vol. 19, No. 1, pp. 1–19. 

Paul Painlev´ e and His Contribution to Science Alexey V. Borisov1, 2, 3* and Nikolay A. Kudryashov4** 1

Institute of Computer Science, Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia 2 A. A. Blagonravov Mechanical Engineering Research Institute of RAS, ul. Bardina 4, Moscow, 117334 Russia 3 Institute of Mathematics and Mechanics of the Ural Branch of RAS, ul. S. Kovalevskoi 16, Ekaterinburg, 620990 Russia 4 Department of Applied Mathematics, National Research Nuclear University “MEPhI”, Kashirskoe sh. 31, Moscow, 115409 Russia Received December 9, 2013; accepted January 10, 2014

Abstract—The life and career of the great French mathematician and politician Paul Painlev´e is described. His contribution to the analytical theory of nonlinear differential equations was significant. The paper outlines the achievements of Paul Painlev´e and his students in the investigation of an interesting class of nonlinear second-order equations and new equations defining a completely new class of special functions, now called the Painlev´e transcendents. The contribution of Paul Painlev´e to the study of algebraic nonintegrability of the N -body problem, his remarkable observations in mechanics, in particular, paradoxes arising in the dynamics of systems with friction, his attempt to create the axiomatics of mechanics and his contribution to gravitation theory are discussed. MSC2010 numbers: 01-00; 01A55; 01A60 DOI: 10.1134/S1560354714010018 Keywords: mathematician, politician, Painlev´e equations; Painlev´e transcendents; Painlev´e paradox

´ 1. THE LIFE AND CAREER OF PAUL PAINLEVE The outstanding French mathematician and politician, Paul Painlev´e was born on December 5, 1863, in Paris. His father, Leon Painlev´e, like his grandfather, was a lithographic draughtsman. Later, when Painlev´e recalled his childhood, he wrote that “he was brought up in the simple democratic atmosphere of French skilled artisan family life”. ´ Paul attended the Ecole Primaire where he showed himself to be extraordinarily gifted in all subjects. But the best progress he made was in mathematics, history and literature. Moreover, he displayed very early a pronounced talent for organization. Paul Painlev´e’s childhood fell on the time of tumultuous political events resulting in the destruction of the French monarchy, which was not restored even after suppression of the Commune of Paris on May 28, 1871. On August 31, 1871, the National Assembly introduced the title of the president of the republic with a three-year term. By the time his secondary education was completed his future was still uncertain. He could not decide whether to take up politics or engineering, but ´ eventually he chose to focus on his research career. He finally entered the Ecole Normale Sup´erieure (now part of the universities of Paris) in 1883 to study mathematics. In 1886 he successfully graduated from it. After Paris, Paul visited the Goettingen university to have the following period of education and prepare his thesis in the complex function theory. He dealt with famous scientists such as the German mathematicians Felix Klein and Herrmann Amandus Schwarz. Both of them had a great impact on Painlev´e’s mind. * **

E-mail: [email protected] E-mail: [email protected]

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Despite the fact that Paul completed his thesis in Germany, he presented it later in Paris, where he became a Doctor of Philosophy. The results of his thesis can be found in [41]. In 1887 Paul Painlev´e began to teach mathematics at Universit´e de Lille. While teaching in Lille, Painlev´e worked on the theory of differential equations and published several papers in scientific journals [42, 43]. Paul Painlev´e taught mathematics at Universit´e de Lille for five years, but in 1892 he returned to Paris where he taught both at the Faculty of Science at the Sorbonne and ´ at the Ecole Polytechnique. This was a rapid return to Paris that showed the high regard in which he was held. A few years later, in 1896, Paul Painlev´e began to teach at the ´ Coll`ege de France and at the Ecole Normale Superieure. As a mathematician Painlev´e was interested at that time in rational transformations of algebraic curves and surfaces [45, 46]. He showed a great interest in the analytical theory of nonlinear equations, which was born in the works of L. Fuchs in the early 1880ies and took shape after an extraordinarily successful work of S. V. Kovalevskaya (1888) on the integration of the problem Fig. 1. Paul Painlev´e. of motion of a rigid body with a fixed point. In 1895 Painlev´e began to study second-order first-degree nonlinear differential equations. He made an attempt at solving the problem formulated by Henri Poincar´e in 1884 in one of his first works — that of determining new special functions as solutions of nonlinear differential equations. This problem was formulated in 1884 by Henri Poincar´e who tried to find new special functions by using first-order equations of arbitrary degree, but failed to find new functions, since all equations which he studied had solutions expressed in terms of the Jacobi elliptic functions. Later, in his evaluation of the scientific contribution of Paul Painlev´e to the theory of differential equations, Jacques Hadamard wrote: “Painlev´e continued the work of Henri Poincar´e, which reached the limit of a man’s power”. No doubt Painlev´e was a distinguished mathematician. In the Encyclopaedia Britannica one can read: “A brilliant and noted French mathematician, Paul Painlev´e broadened the horizons of mathematical knowledge of differential equations. He found new special functions, which are transcendental functions considerably different from the classical special functions”. Painlev´e received many awards for his outstanding mathematical work. Among his awards were the prestigious Prix Bordin of the French Academy of Sciences in 1894. Two years later, in 1896, Painlev´e was awarded the Prix Poncelet. In 1900 he was elected as a member to the Geometry Section of the French Academy of Sciences. Painlev´e married Marguerite Petit de Villeneuve in 1901. Their son Jean was born on November 20, 1902. The son of Paul Painlev´e was a well-known film director, actor, translator, animator, critic and theorist. A few days after his birth, his mother and wife of Paul Painlev´e, Marguerite Petit de Villeneuve, died from complications arising from an infection contracted during childbirth. Painlev´e, the only son, was raised by his father’s sister Marie, a widow. The wish to go into politics, which Paul Painlev´e first conceived in his youth, appeared again when he was forty years old, although there is evidence that he had taken up the cause of Alfred Dreyfus before his active political career, which he began in 1906, when he was elected as a deputy of parliament for the fifth arrondissement of Paris — the Latin Quarter, in which the Universit´e de Sorbonne is situated. In this district there is still a large number of teachers and students. In the parliament Paul Painlev´e very quickly established himself as a head of the Republican-Socialist party. In 1910 he almost stopped his research and teaching activities and ceased to publish his scientific papers. After that Paul Painlev´e was at the head of several parliamentary committees where he competently solved issues of military policy, navy and aviation in the interest of national security of France. Paul Painlev´e had a special interest in aviation, using his theoretical competence in mechanics to study the theory of flight. As a member of parliament, he appealed to the government in 1907 arguing that it was necessary to set up a branch of the military involved with aviation. Painlev´e REGULAR AND CHAOTIC DYNAMICS

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foresaw great possibilities of aviation in military operations. Later his idea was supported in many countries. Airforce still plays a great role in military operations. Painlev´e was Wilbur Wright’s first passenger who made a record 1 hour 10 minute flight in 1908. (In most countries the brothers Wright are recognized to have invented and built the first airplane). In 1909 Painlev´e created the first university course in aeronautical mechanics at Universit´e de Sorbonne. Some historians believe that although Paul Painlev´e achieved the highest posts in his political career, his contribution as a mathematician was much more significant [40]. In their paper about Paul Painlev´e O’Connor and Robertson wrote: “It may seem unfair to say he was less skilled in politics than mathematics when he achieved the highest possible office in politics, but this statement is more meant to comment on his truly outstanding mathematical contributions. Painleve had a naturally simple and unaffected manner, and was possessed of a singular charm that few persons, even among his opponents, were able to resist. His energy was untiring”. Painlev´e died on October 29, 1933, in Paris at the age of 69 and was enterred in the Panth´eon. ´ EQUATIONS 2. THE PAINLEVE One of the greatest achievements made by Paul Painlev´e was the discovery of six nonlinear differential equations with remarkable properties. It is well known that the classical special functions such as Bessel functions, Legendre polynomials, hypergeometric function, Airy function and many others are solutions of linear ordinary differential equations and can be defined as their solutions. One can introduce many of such functions, which are solutions of linear differential equations, including those which are not used in applications and thus are of no interest. The solutions of differential equations can have various types of singular points. In the early 1880ies L. Fuchs noticed that the singular points of general solutions of nonlinear differential equations can depend on the initial conditions. They are called movable singular points and can often be a source of troubles in defining solutions, since such a solution often cannot lead to a onevalued function. The singular points of solutions of linear differential equations are defined only by the form of the equation itself, and this does not cause such difficulties. The above-mentioned special functions are solutions of linear ordinary differential equations of second order of first degree and have fixed (immovable) singular points on a complex plane. This is a general property of all linear differential equations. It should be noted that not all nonlinear differential equations lose this property, which is, generally speaking, typical of linear differential equations. The greatest difficulty in defining the general solution of a nonlinear differential equation is that the solution has movable critical singular points, which raises the question of classification of equations whose solutions have no movable critical singular points in the complex plane. It is this property that is now called the Painlev´e property. Equations that possess the Painlev´e property are said to be Painlev´e integrable. This definition of integrability of nonlinear differential equations is sometimes not satisfied by equations that are integrable by other properties, but in general many equations that are Painlev´e integrable are integrable by other definitions as well. L. Fuchs showed that among all equations of first order of first degree having a polynomial form from a dependent and independent variable, only the general solution of the Riccati equation has no movable critical singular points and, consequently, possesses the Painlev´e property. In the 19th century elliptic functions were discovered which are solutions of nonlinear equations of first order, but of second degree. These elliptic functions extended the family of special functions, they have movable noncritical singular points in the form of poles. An important contribution to the analytical theory of nonlinear differential equations was made by S. V. Kovalevskaya, who was the first to use the Painlev´e property (which had not received this name by that time yet) in the analysis of the general solution of the problem of rotation of a rigid body with a fixed point (the problem of a spinning top). She found all cases where the general solution of this problem had no movable critical singular points. S. V. Kovalevskaya found four cases of parameter values of the problem, where the solutions have no movable critical singular points. Three cases found by S. V. Kovalevskaya were well known from the works of Euler and Lagrange, and the fourth case of parameter values was new (now this case is named after Kovalevskaya). Kovalevskaya integrated a system of equations describing the problem of a top for her case and REGULAR AND CHAOTIC DYNAMICS

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obtained a solution expressed in terms of a hypergeometric function. In 1888 S. V. Kovalevskaya was awarded the Prix Bordin of the French Academy of Sciences for a considerable advance in the solution of the problem of motion of a rigid body with a fixed point. Picard suggested an analysis of a new class of equations — the equation of second order of first degree, which has the form [60]: wzz = R(z, w, wz ),

(2.1)

where R(z, w, wz ) is a rational function in w and wz and is locally analytic in z. It was Paul Painlev´e who set about investigating this class of equations in 1895. To analyze the nonlinear differential equations, Painlev´e devised a new method, called the α-method of Painlev´e [49]. This method consisted of two stages. At the first stage this method allowed one to find necessary conditions for the absence of movable critical singular points. The second stage of the α-method of Painlev´e is extremely tedious. At this stage a direct integration of nonlinear differential equations was performed in each specific case, resulting either in the well-known solution of the equation or in the proof that a new special function arises. The work was very laborious and lasted for many years, from 1895 to 1910 [51–53, 55, 56]. The result of this work was that Painlev´e found fifty second-order nonlinear differential equations of the canonical form (2.1) with general solutions which have no movable critical singular points. The solutions of forty-four equations from this list were obtained in terms of elementary and classical functions, while the solutions of six equations (now called the Painlev´e equations) defined new special functions (called the Painlev´e transcendents). These six equations are presented in the paper [31] included in this issue of the journal. In fact, Paul Painlev´e found only three (P1, P2 and P3) from six equations having the form [8, 15, 29– 31, 52, 53, 55, 56]: (P1 ) (P2 ) (P3 ) (P4 ) (P5 ) (P6 )

yzz = 6y 2 + z,

(2.2)

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yzz = 2 y + z y + α,  1 2 y 2 yz δ + α y + β + γ y3 + , yzz = z − y z z y   3 y2 β yzz = z + y 3 + 4zy 2 + 2 z 2 − α y + , 2y 2 y    2  1 (y − 1) y y (y + 1) yz 1 β 2 + yz − + +γ +δ , yzz = αy + 2 2y y − 1 z z y z y−1     1 1 1 1 1 1 1 2 + + y − + + yz yzz = 2 y y−1 y−z z z z−1 y−z   y (y − 1) (y − z) z−1 z (z − 1) z + +δ α+β 2 +γ . y z 2 (z − 1)2 (y − 1)2 (y − z)2

(2.3) (2.4) (2.5) (2.6)

(2.7)

In this sense Paul Painlev´e never completed the work he had in mind. He got his student Gambier involved in this investigation, and so three remaining equations were found by him [14]. The most general form of the sixth equation, P6, was obtained in 1905 by Richard Fuchs, the son of L. Fuchs [12, 13]. As mentioned above, most equations in the classification performed by Painlev´e and Gambier were reduced to linear differential equations or solved using elliptic functions, and only six equations ultimately turned out to be irreducible to the classical special functions. In fact, during the life of Painlev´e, the irreducibility of his equations was a contentious issue. Without giving any proofs, R. Liouville asserted that the first equation of Painlev´e was reducible to a linear fourth-order equation. Paul Painlev´e disagreed with Liouville and was sure that the equations discovered by him defined new special functions. This problem remained open for many years until the last decade of the 20th century when it was finally solved by the Japanese mathematician Umemura. As expected, all Painlev´e equations REGULAR AND CHAOTIC DYNAMICS

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turned out to be irreducible: they cannot be reduced to linear equations of any order and cannot be reduced to nonlinear first-order equations. The solutions of equations P1 , P2 and P4 are meromorphic functions of an independent variable, but the equations P3 , P5 can be transformed to a form in which the solutions are meromorphic functions. For the equation P6 the points 0, 1 and ∞ are fixed critical points. The equation P6 contains five other equations in the sense that all the other Painlev´e equations can be obtained from it by a passage to the limit. The Painlev´e equations possess a number of interesting properties. A very important property is that all Painlev´e equations have no first integrals in polynomial form. This property implies that these equations define a new type of special functions – the Painlev´e transcendents, which are not the classical transcendental functions. For some parameter values all Painlev´e equations, except for P1 , have a set of rational and special solutions, which can be found by means of recurrence relations. For a long time, the Painlev´e equations were regarded as representatives of only a certain class of equations with solutions that have no movable critical points. It was not until the 1970ies that numerous applications of the Painlev´e equations were described in the literature, which is the major reason for ongoing intensive research into these equations. More recently a great deal of effort has gone into the study of higher analogs of the Painlev´e equations, which were introduced and partially explored at the end of the last century in [23–28]. ´ PROPERTY, BRANCHING OF SOLUTIONS 3. THE PAINLEVE AND NONEXISTENCE OF FIRST INTEGRALS Using the ideas of K. Weierstrass, L. Fuchs and S. V. Kovalevskaya, P. Painlev´e studied nonlinear differential equations relative to the behavior of the solution near the singularities of movable singular points and its branching on the complex time plane. This property is often called the Painlev´e property (or the Painlev´e test), primarily in the physics literature. There exist several types of this property, in particular, what Painlev´e studied is often called at present the strong Painlev´e property. At the necessary number of arbitrary constants the expansion of the solution of the equation in a Puiseux series is often called the weak Painlev´e property. If we restrict ourselves to the parameters of the problem, we have the conditional Painlev´e property. It should be emphasized that probably it would be more correct to call the Painlev´e property the Kovalevskaya property, since it was S. V. Kovalevskaya who, based on the ideas of K. Weierstrass and L. Fuchs, was the first to pose the problem of a full parametric family of meromorphic (in the physical terminology, the strong Painlev´e property) solutions for the Euler – Poisson equations. According to her calculations, this should yield a single-valued general solution in the form of theta-functions. The idea turned out to be fruitful, and afterwards she succeeded in finding a new integrable case and the well-known Kovalevskaya integral, which — unlike the Euler and Lagrange cases found earlier — turned out to be nontrivial and of the fourth degree in momenta. After these results had been obtained, a natural question arose as to the relation between the Painlev´e property and the integrability of the system by quadratures. By the way, it was Painlev´e [51] who noticed the relation between the Painlev´e property and the integrability of equations, therefore, at present the equations having the Painlev´e property are said to be integrable in the Painlev´e sense. However, one should bear in mind that the Painlev´e property is only a necessary but insufficient condition for integrability. In [34] A. M. Lyapunov modified and refined the analysis of S. V. Kovalevskaya (in response to the objections of academician A. Markov) and introduced equations in variations near general meromorphic solutions. This idea was also productive and ultimately led S. L. Ziglin [69] to general results about the branching of solutions on the complex time plane and about the existence of complex meromorphic first integrals. Formalizing the ideas of S. L. Ziglin and using the ideas of differential Galois theory, J. J. Morales-Ruiz and J.-P. Ramis developed a separate research field. The current status of this field and an overview of solved problems can be found in [38, 39]. An alternative approach to the Painlev´e problem was developed by V. V. Kozlov [19]. It implies that to analyze the branching of Hamiltonian systems, it is necessary to use action-angle variables. We also mention the paper by H. Yoshida [68], where he introduced the Kovalevskaya exponents, REGULAR AND CHAOTIC DYNAMICS

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which were related to the degrees of homogeneity of additional first integrals of quasi-homogeneous systems (see also [20]). The problem of polynomial integrals of Hamiltonian systems and branching of solutions on the complex time plane was formulated in [21]. 4. ALGEBRAIC NONINTEGRABILITY OF THE N -BODY PROBLEM In [50] Painlev´e strengthened the result of Bruns on the nonexistence of algebraic integrals of the N -body problem — a system of material points subject to mutual attraction by the Newton gravitation law. In contrast to Bruns, Painlev´e used first integrals which are algebraic only in the velocities of material points, but are arbitrary relative to coordinates. Physically such a problem statement has much more sense, since all well-known integrals in Hamiltonian dynamics are polynomial in the velocities, and the algebraic dependence on the coordinates in the Bruns theorem is invariant under arbitrary analytical changes of coordinates (as a consequence, Wintner [66] believed that the Bruns theorem has nothing to do with dynamics). For the current status of the issue of existence of first integrals of the N -body problem (the main advances here are associated with the Morales-Ramis technique) see also the overview [39]. 5. SOME REMARKABLE OBSERVATIONS IN MECHANICS We also note two examples given by P. Painlev´e, which have been used in the scientific and teaching literature. One of them is related to the Lagrange – Dirichlet theorem; when the necessity of the condition of strict local minimum of the potential is illustrated, an infinitely differentiable function is usually used as an example: U (q) = (cos q −1 ) exp(−q −2 ),

q = 0,

U (0) = 0,

for which the point q = 0 is not a point of strict local minimum (the example of Painlev´e – Wintner → Arnold, Kozlov, Neishtadt). Another example is the problem of motion of a heavy material point on the surface of a paraboloid. In [47], P. Painlev´e found an additional first integral and thus showed the possibility of integrating by quadratures. This problem was integrated using the method of separation of variables by S. A. Chaplygin [7], who also described geometrically various trajectories of the system. A nonholonomic generalization of the Painlev´e – Chaplygin integral was proposed in [4]. 6. DYNAMICS OF SYSTEMS WITH FRICTION AND PARADOXES Another line of investigations pursued by P. Painlev´e, which still gives rise to a steady flow of new investigations, is associated with friction. In 1895 two courses of Painlev´e’s lectures were published: “Le¸cons sur l’int´egration des ´equations diff´erentielles de la m´ecanique et applications” [47] and “Le¸cons sur le frottement” [48]. The former only touched upon, while the latter systematically analyzed motions of rigid body systems, which involve the force of dry (Coulomb) friction during contact (sliding). Painlev´e gave general equations of motion of such systems and noted paradoxical situations to which the use of the Coulomb friction law may lead. In the statics such paradoxes were first noted by the well-known Irish mecanic Jellett, who in “A Treatise on the Theory of friction” [17] wrote that such paradoxes cannot occur in dynamics. This assertion was not confirmed afterwards. These lectures [47, 48] of Painlev´e and his subsequent works [54] gave rise to the well-known controversy between some renowned mecanics such as L. Lecornu [32], de Sparre [9], F. Klein [18], R. Von Mises [37], G. Hamel [16], and L. Prandtl [62]. Without going into details of this controversy, during which various modifications of the Coulomb law were proposed, we note that it ended in a general understanding of the necessity to take into account the elasticity in the analysis of motion of such systems and in a further experimental study of the laws of dry friction. Figures 2 and 3 show two examples of systems with Coulomb friction: Painlev´e – Klein [18, 47, 59] and Painlev´e – Appel [2, 9, 47], which were examined in the greatest detail during that controversy. Depending on the initial conditions, both nonuniqueness of solutions and their absence is possible in these systems during sliding. The first of experimental devices for experimental investigations of the dynamics of the Painlev´e – Klein system was designed by REGULAR AND CHAOTIC DYNAMICS

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Fig. 2. Example of Painlev´e – Klein: two material points M and M1 of unit mass are connected by means of a rigid weightless rod M M1 and slide with friction along two parallel guide rails. Constant active forces X1 , X2 , parallel to the axis OX, are applied to these points.

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Fig. 3. Example of Painlev´e – Appel: two material points M and M1 of unit mass are connected by means of a rigid weightless rod M M1 . The point M can slide along a rough straight line OX that has the friction coefficient μ.

L. Prandtl and described by F. Pfeiffer in [59]. For information on recent experiments and theoretical investigations see [64]. At present, however, most questions of the behavior of the system in paradoxical situations discovered by Painlev´e remain open and require an additional study. Some of these questions are covered in other papers of this issue of RCD. After the controversy (which came to an end in 1930) the interest in the Painlev´e paradoxes subsided until it was revived in the last decades, when a new surge of investigations arose, inspired mainly by the development of methods of nonlinear dynamics, the theory of bifurcations of nonsmooth systems, the theory of differential inclusions, linear programming, and by the discussion of paradoxical situations involving dry friction in more complex mechanical systems important for applications (for example, robotics). 7. THE AXIOMATICS OF MECHANICS Another remarkable book by Paul Painlev´e published in 1909, is devoted to the axioms of mechanics [57]. He believed that the axioms of mechanics allow a definition of the absolute coordinate system of motions only for straight-line and uniform translational motion. In his opinion, such an uncertainty does not create any inconveniences in solving the problems in mechanics, i. e., by choosing one of the absolute coordinate systems it is straightforward to prove that the absolute forces defined above do not change. In no way did Painlev´e question the existence of absolute space. Poincar´e [61], on the contrary, rejected the idea of absolute space and asserted a relative nature of any motion. In later works, such as [22, 35, 58], all motions were presented as relative, and the inertial reference frames (relative to which the motion of a material point that is not acted upon by any forces is rectilinear and uniform) are considered to be equivalent. ´ HYPOTHESIS AND COLLISIONLESS SINGULARITIES 8. PAINLEVE’S An important aspect of Painlev´e’s research efforts aimed at the study of differential equations was the investigation of singularities of the N -body problem. As is well known, the Newtonian potential has a singularity in the zero and not every solution can be extended to the entire time axis [0, t∗ ), where 0 < t∗ < f y, t∗ is called the singularity of this solution. There exists a close relation between the singularities of the system and the singularities of solutions. In 1895 King Oscar II of Sweden and Norway invited Paul Painlev´e to give lectures at the University of Stockholm, which are presented in [47]. The king attended the first of them. In these lectures Painlev´e placed particular emphasis on singularities of the N -body problem. In 1897 he also published the lectures [49], which contained the first systematical study of these singularities. Let p(q) = min |qi − qj |, |qi − qj | be 1i
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the distance between material points. In 1895 Painlev´e showed that for an analytical solution of   ∗ ∗ the N -body problem, defined on [0, t ), t is a singularity if and only if lim∗ ρ q(t) = 0. t→t

However, not all singularities end in a collision. There may exist singularities without collision, for example, they have been found in the two-body problem. In 1895 P. Painlev´e also showed that in the three-body problem all singularities arise due to collisions. He failed to carry over his result to N  4 and advanced a hypothesis that for N  4 there exist solutions of the N -body problem which are a singularity without collision. A proof of validity of Painlev´e’s hypothesis was obtained almost hundred years later. In 1975 J. N. Mather and R. McGehee [36] gave an example of pseudocollision in an additional four-body problem. In one end position two material points form a cluster, in another end position a cluster arises consisting of one material point, and the remaining material point oscillates between two clusters. In this example, however, pseudo-collision was not the first of the arising singularities, as P. Painlev´e assumed. It was not until 1987 that Z. Xia [67] published a complete confirmation of Painlev´e’s hypothesis. His example is associated with the spatial five-body problem, in which two pairs of bodies form clusters and the fifth body oscillates between them. For the current status of the problem of collisions see [63] and [10, 11]. 9. TRANSFORMATION OF DYNAMICAL EQUATIONS AND GEODESIC EQUIVALENT METRICS In 1982–1894 Painlev´e published a series of studies [44, 45] on the transformation of dynamical equations, in which he developed the ideas of Goursat, Appel and Darboux and formulated a number of general propositions of how the form of equations of motion changes if not only the phase variables but also time variables are transformed. Such ideas are used nowadays, for example, to find trajectory isomorphisms of integrable systems [3, 5], and for Hamiltonization of nonholonomic systems (for example, as in Chaplygin [6]). We also note that these works of Painlev´e are purely analytical — at present all his arguments are much easier to understand using geometric ideas — these works led to the general theory of geodesic equivalent flows (metrics) [65] and to projective dynamics [1]. ´ AND GRAVITATION 10. PAINLEVE Paul Painlev´e made an important contribution to general relativity theory. In 1921 he found a solution of Einstein’s equations for a spherically symmetric case in which there is no coordinate singularity on the event horizon, which is present in the Schwarzschild solution. Painlev´e presented his solution in a letter to Einstein, in which he also invited him to Paris to take part in a discussion. In his reply (of December 7), Einstein apologized for not being able to come in the nearest future and explained why Painlev´e’s arguments, critical remarks and coordinates did not satisfy him. Finally, in early April Einstein arrived in Paris. On April 5, 1922, during a discussion at Coll`ege de France (in which Painlev´e, Becquerel, Brillouin, Cartan, de Donder, Hadamard, Langevin and Nordmann participated and which was devoted to singular potentials) Einstein, perplexed by the presence of a nondiagonal term in an interval, rejected Painlev´e’s solution. Thus, Painlev´e’s proposal was abandoned for a long time. Later, in 1933, a relation between the solutions of Schwarzschild and Painlev´e was found by Lemaitre [33]. Let us consider this relation in more detail. The interval in the Schwarzschild metric is defined by the expression:   2M dr 2 dt2 − − r 2 dθ 2 − r 2 sin2 θ dϕ2 , ds2 = 1 − 2M r 1− r where t, r, θ, ϕ are the Schwarzschild coordinates, M is the mass of an object generating a gravitational field. Note that in the chosen system of units the velocity of light and the gravitation constant are equal to unity (G = 1, c = 1). As we see, the above metric has a singularity at r → 2M . We show that this singularity is not physical and can be eliminated by a passage to the Gullstrand – Painlev´e coordinates, tr , r, θ, ϕ:  1 df 2M = . tr = t − f (r), 2M dr r 1− r REGULAR AND CHAOTIC DYNAMICS

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Fig. 4. The dispute between Albert Einstein and Paul Painleve during Einstein’s visit in Paris.

In the new coordinates the Schwarzschild metric passes into the solution found by Painlev´e:    2M 2M 2 2 dtr − 2 dtr dr − dr 2 − r 2 dθ 2 − r 2 sin2 θ dϕ2 . ds = 1 − r r As we see, there is no singularity at r → 2M in the Painlev´e solution. ´ 11. POLITICAL ACTIVITIES OF PAUL PAINLEVE From October 29, 1915 to December 12, 1916, Paul Painlev´e served as Minister for Public Instruction and Inventions at the government of Aristide Briand (March 28, 1862 – March 7, 1932). On November 15, 1915, in addition to his first ministerial post, Painlev´e also became Minister of Inventions for National Defence, undertaking great efforts for coordination of scientific research in the interest of the military department. From March 20, 1917 to September 7, 1917, Paul Painlev´e was a War Minister at the government of Alexandre F´elix Joseph Ribot (February 7, 1842 – January 13, 1923). In September 1917 Painlev´e succeeded Ribot as prime minister, who had lost the support of the Socialists and resigned. Paul Painlev´e’s tenure as prime minister lasted only until November 17, 1917. Painlev´e played a leading role in the Rapallo conference convened by the Entente Powers in 1917. After his resignation, Painlev´e did not play any notable role in the political life of France until the election which took place in November 1919, when he emerged as a critic of the government. ´ By the time the next election took place in May 1924, Paul Painlev´e’s collaboration with Edouard Herriot (July 5, 1872 – March 26, 1957) had led to the formation of the Cartel des Gauches, which won the election. The government of Briand resigned, and Herriot was appointed Prime Minister on June 14, 1924, while Paul Painlev´e became President of the Chamber of Deputies. ´ Painlev´e remained President of the Chamber of Deputies until April 1925. On April 17 Edouard Herriot resigned due to a financial crisis, and the President of France appointed Paul Painlev´e prime minister again. Painlev´e held this post until November 28, 1925. However, both the second and the third governments of Paul Painlev´e were weak. On October 29, 1925, the second government, at which Painlev´e was War Minister, fell, but Painlev´e remained Prime Minister. In addition, he became Minister of Finance and held this post until November 1925. However, due to problems of French troops in Syria and due to a financial crisis in France, Painlev´e resigned on November 21, 1925. On November 28, 1925, the third government of Paul Painlev´e was dissolved, and Aristide Briand became Prime Minister again. At the ninth government of Briand, which existed until July 19, 1926, Paul Painlev´e held the post of a war minister until June 23, 1926. Paul Painlev´e was appointed War Minister again at the eleventh government of Briand on July 29, 1929, and held this post until November 3, 1929. REGULAR AND CHAOTIC DYNAMICS

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Fig. 5. Field Marshal Sir Douglas Haig and Paul Painleve in 1917.

Paul Painlev´e was Minister of Air at the later governments: at the government of Th´eodore Steeg (December 18, 1868 – December 19, 1950), which existed from December 13, 1930, through January 27, 1931, and at the government of Joseph Paul-Boncour who served as Prime Minister from December 18, 1932, to January 31, 1933. As Minister of Air he made proposals for an international agreement to end the production of bombers in all countries, but his proposals were not supported. In May 1932, Paul Painlev´e was proposed for President of France, but he withdrew before voting took place. The resignation of the government of Paul-Boncour in January 1933 put an end to the political career of Paul Painlev´e. ACKNOWLEDGEMENTS The authors warmly thank Alain Albouy, Igor V. Volovich, Tatyana B. Ivanova and Ivan A. Bizyaev for their helpful suggestions and valuable advice on the interpretation of Painlev´e’s works and his contribution to various areas of mathematics.

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´ IN CHRONOLOGICAL ORDER1) LIST OF SCIENTIFIC WORKS BY PAUL PAINLEVE 1886 1. Sur le d´eveloppement en s´erie de polynˆomes d’une fonction holomorphe dans une aire quelconque. (C.R. 102; 672–675). 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

1887 Sur les ´equations lin´eaires simultan´ees aux d´eriv´ees partielles. (C.R.2) 104; 1497–1501). Sur les lignes singuli`eres des fonctions analytiques (Th`eses pr´esent´ees `a la Fac. des Sciences de Paris 1886–87). (Ann. Fac. Sci. Univ. Toulouse 2; 130 pages). Sur les ´equations diff´erentielles lin´eaires du troisi`eme ordre. (C.R. 104; 1829–1832). Sur les ´equations diff´erentielles lin´eaires. (C.R. 105; 58–61). Sur les transformations rationnelles des courbes alg´ebriques. (C.R. 105; 792–794). Sur Sur Sur Sur Sur

la repr´esentation conforme les ´equations diff´erentielles les ´equations diff´erentielles les ´equations diff´erentielles les ´equations diff´erentielles

1888 des polygones. (C.R. 106; 473–476). lin´eaires `a coefficients alg´ebriques. (C.R. 106; 535–537). du premier ordre. (C.R. 107; 221–224). du premier ordre. (C.R. 107; 320–323). du premier ordre. (C.R. 107; 724–726).

1889 12. Sur la transformation des fonctions harmoniques et les syst`emes triples de surfaces orthogonales. (Travaux et M´emoires des Facult´es de Lille, 1; 1–29; aoˆ ut 1889). 13. 14. 15. 16. 17.

18. 19. 20. 21. 22. 23.

1890 Sur les int´egrales rationnelles des ´equations du premier ordre. (C.R. 110; 34–36). Sur les transformations simplement rationnelles des surfaces alg´ebriques. (C.R. 110; 184–186). Sur les transformations simplement rationnelles des surfaces et sur une classe d’´equations diff´erentielles. (CR. 110; 226–229). Sur une transformation des ´equations diff´erentielles du premier ordre. (C.R. 110; 840–843). Sur les int´egrales alg´ebriques des ´equations diff´erentielles du premier ordre. (C.R. 110; 945– 948). 1891 Sur les ´equations diff´erentielles du premier ordre (1e partie). (Ann. Sc. Ec. Norm. (3), 8; 9–58 et 103–140). Sur la th´eorie de la repr´esentation conforme. (C.R. 112; 653–657). Sur l’int´egration alg´ebrique des ´equations diff´erentielles du premier ordre. (C.R. 112; 1190– 1193). Sur les ´equations diff´erentielles du premier ordre (2e partie) (Ann. Sc. Ec. Norm. (3), 8; 201–226, 267–284 et 9; 9–30). Remarque sur une communication de M. Markoff (Sur les ´equations diff´erentielles lin´eaires C.R. 113; 685–688), relative a` des ´equations diff´erentielles lin´eaires. (C.R. 113; 739–740). M´emoire sur les ´equations diff´erentielles du premier ordre. (Ann. Sc. Ec. Norm. (3), 8; 9–58).

´ Based on the list given in Painlev´e, P., Œuvres de Paul Painlev´e. Tome I, Paris: Editions du CNRS, 1972, pp. 25–31. 2) C. R. — Comptes rendus de l’Acad´emie des Sciences. 1)

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24. 25. 26. 27. 28. 29. 30. 31. 32. 33.

34. 35. 36. 37. 38. 39. 40. 41.

42. 43. 44. 45. 46. 47. 48. 49. 3)

1892 Sur les int´egrales des ´equations diff´erentielles du premier ordre, poss´edant un nombre limit´e de valeurs. (C.R. 114; 107–109). Sur les int´egrales des ´equations du premier ordre qui n’admettent qu’un nombre fini de valeurs. (C.R. 114; 280–283). Sur les transformations en M´ecanique. (C.R. 114; 901–904). Sur les ´equations diff´erentielles du premier ordre (3e partie). (Ann. Sc. Ec. Norm. (3), 9; 101–144 et 283–308). Sur les transformations en M´ecanique. (C.R. 114; 1104–1107). Sur les int´egrales de la Dynamique. (C.R. 114; 1168–1171). Sur les groupes discontinus de substitutions non lin´eaires `a une variable. (C.R. 114; 1345– 1348). Sur les transformations en M´ecanique. (C.R. 114; 1412–1414). Sur les transformations des ´equations de Lagrange. (C.R. 115; 495–498). Sur la transformation des ´equations de la dynamique. (C.R. 115; 714–717). Rectification d’une faute d’impression (C.R. 115; 874–875). 1893 Sur les mouvements de syst`emes dont les trajectoires admettent une transformation infinit´esimale. (C.R. 116; 21–24). Sur les ´equations diff´erentielles d’ordre sup´erieur dont l’int´egrale n’admet qu’un nombre fini de d´eterminations. (C.R. 116; 88–91). Sur les ´equations diff´erentielles d’ordre sup´erieur dont l’int´egrale n’admet qu’un nombre donn´e de d´eterminations. (C.R. 116; 173–176). Sur les singularit´es essentielles des ´equations diff´erentielles d’ordre sup´erieur. (C.R. 116; 362– 364). Remarque de M. Picard sur cette note (C.R. 116, 365). Sur les transcendantes d´efinies par les ´equations diff´erentielles du second ordre. (C.R. 116; 566–569). Sur les ´equations du second degre dont l’int´egrate g´en´erale est uniforme. (C.R. 117; 211– 214)3) . Sur les ´equations du second ordre a` points critiques fixes et sur la correspondance univoque entre deux surfaces. (C.R. 117; 611–614). Sur les ´equations diff´erentielles du second ordre `a points critiques fixes. (C.R. 117; 686–688). 1894 M´emoire sur la transformation des ´equations de la Dynamique. (J. Math. pures appl. (4) 10; 5–92). Sur une application de la th´eorie des groupes continus `a la th´eorie des fonctions. (C.R. 118; 845–848). Le¸cons sur l’integration des ´equations de la Dynamique et applications. (Paris — Hermann). Sur l’int´egration alg´ebrique des ´equations diff´erentielles lin´eaires. (C.R. 119; 37–40). Note sur un M´emoire de M. Humbert. (J. Math. pures appl. (4) 10; 203–206). Sur une certaine identit´e entre d´eterminanta. (Bull. Soc. Math. France, 22; 116–119). Sur les transformations infinit´esimales des trajectories des syst`emes. (C.R. 119; 637–639). Sur les mouvements et les trajectoires r´eels des syst`emes. (Bull. Soc Math. France, 22; 136– 184).

P.P. avait probablement titr´e sa Note: “Sur les ´equations du second ordre et du premier degr´e. . . ”. REGULAR AND CHAOTIC DYNAMICS

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50. 51. 52. 53. 54.

55. 56. 57. 58. 59. 60. 61. 62. 63.

64. 65. 66. 67. 68. 69. 70. 71.

72. 73. 74. 75. 76. 77.

13

1895 Sur la d´efinition g´en´erale du frottement. (C.R. 120; 596–599). Sur les lois du frottement de glissement. (C.R. 121; 112–115). Sur les surfaces alg´ebriques qui admettent un groupe continu de transformations birationnelles. (C.R. 121; 318–321) Le¸cons sur le frottement. (Paris Hermann). [Traduction Russe en 1954. Moscou]. Painlev´e, P., Le¸cons sur l’int´egration des ´equations diff´erentielles de la m´ecanique et applications, Paris: Hermann, 1895 1896 Sur les fonctions uniformes d´efinies par l’inversion de diff´erentielles totales. (C.R. 122; 660– 662). Sur l’inversion des syst`emes de diff´erentielles totales. (C.R. 122; 769–772). Sur les transformations biuniformes des surfaces alg´ebriques. (C.R. 122; 874–877). Sur les ´equations diff´erentielles du premier ordre. (C.R. 122; 1319–1322). Sur les ´equations diff´erentielles du premier ordre. (R´eponse `a M. Korkine) (C.R. 123; 88–91). Sur les transformations des ´equations de la Dynamique. (C.R. 123; 392–395). M´emoire sur les ´equations diff´erentielles du premier ordre dont l’int´egrale est de la forme h(x)[y − g1 (x)][y − g2 (x)] . . . [y − gn (x)] = c. (Ann. Fac. Sci. Univ. Toulouse (1896)). Sur les singularit´es des ´equations de la Dynamique. (C.R. 123; 636–639). Sur les singularit´es des ´equations de la Dynamique et sur le probl`eme des trois corps. (C.R. 123; 871–873). 1897 Sur les int´egrales premieres des syst`emes diff´erentiels. (C.R. 124; 136–139). Le¸cons sur la th´eorie analytique des ´equations diff´erentielles, profess´ees `a Stockholm (Sept. oct. nov. 1895) sur l’invitation de S. M. le Roi de Su`ede et de Norv`ege, (Paris-Hermann). Sur les int´egrales premi`eres de la Dynamique et sur le probl`eme des n corps. (C.R. 124; 173–176). Sur les int´egrales quadratiques des ´equations de la Dynamique. (C.R. 124; 221–224, Additions C.R. 125; 156). Sur les petits mouvements p´eriodiques des syst`emes. (C.R. 124; 1222–1225). Sur les petits mouvements p´eriodiques des syst`emes `a longue p´eriode. (C.R. 124; 1340–1342). Sur les positions d’´equilibre instable. (C.R. 125; 1021–1024). Sur les cas du probl`eme des trois corps (et des n corps) o` u deux des corps se choquent au bout d’un temps fini. (C.R. 125; 1078–1081). 1898 Sur la repr´esentation des fonctions analytiques uniformes. (C.R. 126; 200–202). Sur le d´eveloppement des fonctions uniformes ou holomorphes dans un domaine quelconque. (C.R. 126; 318–321). Sur le d´eveloppement des fonctions analytiques pour les valeurs r´eelles des variables. (C.R. 126; 385–388). Sur le d´eveloppement des fonctions r´eelles non analytiques. (C.R. 126; 459–461). M´emoire sur les int´egrales premi`eres du probl`eme des n corps. (Bull. astronomique 15; 81– 113). Sur les surfaces qui admettent un groupe infini discontinu de transformations birationnelles. (C.R. 126; 512–515).

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78. Sur les ´equations diff´erentielles du second ordre `a points critiques fixes. (C.R. 126; 1185– 1188). 79. Sur la d´etermination explicite des ´equations diff´erentielles du second ordre `a points critiques fixes. (C.R. 126; 1329–1332). 80. Sur les ´equations diff´erentielles du second ordre `a points critiques fixes. (C.R. 126; 1697– 1700). 81. Sur les ´equations diff´erentielles du second ordre `a points critiques fixes. (C.R. 127; 541–544). 82. Sur les ´equations diff´erentielles du second ordre `a points critiques fixes. (C.R. 127; 945–948).

83. 84. 85. 86. 87. 88. 89. 90. 91.

92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102.

1899 Sur le d´eveloppement d’une branche uniforme de fonction analytique. (C.R. 128; 1277–1280). Sur le calcul des int´egrales des ´equations diff´erentielles par la m´ethode de Cauchy – Lipchitz. (C.R. 128; 1505–1508). Sur le calcul des int´egrales des ´equations diff´erentielles par la m´ethode de Cauchy – Lipchitz. (Bull Soc. Math. France, 27; 149–152). Sur le d´eveloppement d’une branche uniforme de fonction analytique en s´erie de polynˆomes. (C.R. 129; 27–31). Sur le d´eveloppement des fonctions analytiques de plusieurs variables. (C.R. 129; 92–95). Sur les ´equations du second ordre `a points critiques fixes. (C.R. 129; 750–753). Sur les ´equations diff´erentielles du second ordre `a points critiques fixes. (C.R. 129; 949–952). Sur la repr´esentation des fonctions elliptiques. (Bull. Soc. Math. France 27; 300–302). Gew¨ohnliche Differentialgleichungen, Existenz der L¨ osungen. (Encyklop¨ adie der mathematischen Wissenschaften 2.1.1.; 189–229). En fran¸cais: Existence de l’int´egrale g´en´erale. D´etermination d’une int´egrale particuli`ere par ses valeurs initiales. (Encyclop´edie des Sciences Math´ematiques 2.3; 1–57). 1900 Sur les syst`emes diff´erentiels `a points critiques fixes. (C.R. 130; 767–770). Sur les ´equations diff´erentielles du troisi`eme ordre `a points critiques fixes. (C.R. 130; 879– 882). Sur les ´equations diff´erentielles d’ordre quelconque a` points critiques fixes. (C.R. 130; 1112– 1115). Sur une relation entre la th´eorie des groupes continus et les ´equations diff´erentielles `a points critiques fixes. (C.R. 130; 1171–1173). Sur les int´egrales uniformes du probl`eme des n corps. (C.R. 130; 1699–1701). Sur la d´etermination unique de l’int´egrale d’une ´equation diff´erentielle par les conditions initiales de Cauchy. (Bull. Soc. Math. France, 28; 191–196). M´emoire sur les ´equations diff´erentielles dont l’int´egrale g´en´erale est uniforme. (Bull. Soc. Math. France, 28; 201–261). Sur les singularit´es des fonctions analytiques et en particulier, des fonctions d´efinies par les ´equations diff´erentielles. (C.R. 131; 489–492). Sur les syst`emes diff´erentiels `a int´egrale g´en´erale uniforme. (C.R. 131; 497–499). Errata a` cette note (C.R. 131; 534). Sur les ´equations diff´erentielles du second ordre et d’ordre sup´erieur dont l’int´egrale g´en´erale est uniforme. (Acta Math. 25; 1–85). [parution 1902]. Notice sur les travaux scientifiques. Gauthier-Villars.

1901 103. Sur les singularit´es essentielles des ´equations diff´erentielles. (C.R. 133; 910–913). REGULAR AND CHAOTIC DYNAMICS

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1902 104. Remarque sur une communication de P. Boutroux. (Sur la croissance des fonctions enti`eres, C.R. 134; 153–155). (C.R. 134; 155–157). 105. Sur les transcendantes m´eromorphes d´efinies par les ´equations diff´erentielles du second ordre. (C.R. 134; 449–453). 106. Sur le th´eor`eme fondamental de la th´eorie des fonctions ab´eliennes. (C.R. 134; 808–813). 107. Sur le d´eveloppement des fonctions analytiques en s´erie de polynˆomes. (C.R. 135; 11–15). 108. Observations sur la Communication pr´ec´edente (communication de E. Borel: C.R. 135; 150– 152). Sur la g´en´eralisation du prolongement analytique. (C.R. 135; 152–153). 109. Sur les fonctions qui admettent un th´eor`eme d’addition, (Acta Math. 27; 1–54) [parution 1903]. 110. Sur l’irr´eductibilit´e des transcendantes uniformes d´efinies par les ´equations diff´erentielles du second ordre. (C.R. 135; 411–415). 111. D´emonstration de l’irr´eductibilit´e absolue de l’´equation, y  = 6y 2 + x (C.R. 135; 641–647). 112. Sur les transcendantes uniformes d´efinies par l’´equation, y  = 6y 2 + x (C.R. 135; 757–761). 113. Sur l’irr´eductibilit´e de l’´equation, y  = 6y 2 + x (C.R. 135; 1020–1025). [113a]. Rapport du Grand Prix des Sciences Math´ematiques [Vessiot]. (C.R. 137; 1154–1162). 114. Sur les ´equations diff´erentielles du second ordre et d’ordre sup´erieur dont l’int´egrale g´en´erale est uniforme, (Acta Math. 25; 1–85). 1903 115. Sur la r´eductibilit´e des ´equations diff´erentielles. (C.R. 136; 189–193). 1904 116. Sur la stabilit´e de l’´equilibre. (C.R. 138; 1555–1557). 117. Sur le th´eor`eme des aires et les syst`emes conservatifs. (C.R. 139; 1170–1174). [117a]. Paroles et Ecrits (3). 1905 118. Sur les lois du frottement de glissement. (C.R. 140; 702–707). 119. Sur les lois du frottement de glissement. (C.R. 141; 401–405). 120. Sur les lois du frottement de glissement. (C.R. 141; 546–552). ´ Le¸cons sur les 121. Painlev´e, P., Sur le d´eveloppement des fonctions analytiques, in Borel, E., fonctions de variables r´eelles et les d´eveloppements en s´erie de polynˆ omes, Paris: GauthierVillars, 1905, pp. 101–147. 1906 122. Sur les equations differentielles du second ordre a` points critiques fixes. (C.R. 143; 1111– 1117). 1907 123. Rapport sur le prix Vaillant [d´ecern´e `a J. HADAMARD]. (C.R. 145; 983–986). 1908 124. Sur les ´equations diff´erentielles du premier ordre dont l’int´egrale g´en´erale n’a qu’un nombre fini de branches. (Note figurant a` la fin d’un ouvrage de P. Boutroux: Le¸cons sur les fonctions d´efinies par les ´equations diff´erentielles du 1er ordre. (Cours profess´e au Coll`ege de France)). Gauthier-Villars. REGULAR AND CHAOTIC DYNAMICS

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1909 125. Observations au sujet de la communication pr´ec´edente (Sur les fonctions analytiques uniformes qui restent continues sur un ensemble parfait discontinu de singularit´es. Note de M. Denjoy C.R. 148; 1154–1156) (C.R. 148; 1156–1157). 1910 126. avec E. Borel. L’Aviation (Alcan) (Voir “Paroles et Ecrits” (14) et (14 bis)). 127. Aviacado ed aeroplane. Tradukita en id. da L. Cuesnet. Paris — Librairie a´eronautique. 128. Existence de l’int´egrale g´en´erale, in Encyclop´edie des Sciences math´ematiques, t. II, vol 3, Paris, 1910, fasc. l, pp. 1–57. 1911 ´ 129. Etude sur le r´egime normal d’un a´eroplane. (La Technique A´eronautique 1; 3–11). 1912 130. Rapport sur le Grand Prix des Sciences Math´ematiques. [Boutroux, Chazy, Gamier]. (C.R. 155, 1284–1291). 19214) 131. Note sur la communication pr´ec´edente. (Note de M. Pescara, 845–847: R´esultat sur les essais r´ecents d’un h´elicopt`ere). (C.R. 172; 847–848). 132. La M´ecanique classique et la th´eorie de la relativit´e. (C.R. 172; 677–680). 133. La gravitation dans la M´ecanique de Newton et dans la M´ecanique d’Einstein. (C.R. 173; 873–887). 1922 134. La th´eorie classique et la th´eorie einsteinienne de la gravitation. (C.R. 174; 1137–1143). 135. Note sur les deux communications pr´ec´edentes. (Note de Chazy; sur les v´erifications astronomiques de la th´eorie de la relativit´e (mˆeme tome p. 1157–1160). Note de Trousset: Les lois de K´epler et les orbites relativistes (mˆeme tome p. 1160–1161). (C.R. 174; 1161–1162). 136. Les axiomes de la M´ecanique. Examen critique. Gauthier-Villars. (nouveau tirage 1955). 137. La m´ecanique classique et la th´eorie de la relativit´e. (L’Astronomie 36; 6–9). 1927 138. Les r´esistances d’un liquide au mouvement d’un solide. (J. Ec. Poly. (2) 26. 165–182). 1928 139. Sur le d´eveloppement des fonctions analytiques. Note figurant dans l’ouvrage d’Emile Borel: Le¸cons sur les fonctions de variables r´eelles et les d´eveloppements en s´erie de polynˆomes. Gauthier-Villars et Cie. 1929 140. Cours de M´ecanique — M´ecanique des solides ind´eformables. M´ecanique des milieux continus. Th´eorie des machines et aviation. M´ecanique de Newton et d’Einstein. P. Painlev´e et Platrier. Gauthier-Villars. 1930 ´ 141. Cours de m´ecanique (Ecole polytechnique). Gauthier-Villars. 142. R´esistance des fluides non visqueux. Gauthier-Villars. 1961 143. Traynard: Fonctions ab´eliennes et fonctions th´eta de deux variables d’apr`es un cours de P. Painlev´e. (Memorial Sc. Math.). ´ 144. TISSERAND: Recueil compl´ementaire d’exercices d’Analyse. Edition augmentee par ´ PAINLEVE. Paris (1896). 4)

Le nom de P. Painlev´e n’apparait pas dans la correspondance A. Einstein, M. Born [le Seuil 1972]. REGULAR AND CHAOTIC DYNAMICS

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