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Regular and Chaotic Dynamics

, Volume 19, Issue 1, pp 1–19 | Cite as

Paul Painlevé and his contribution to science

  • Alexey V. Borisov
  • Nikolay A. Kudryashov
Article

Abstract

The life and career of the great French mathematician and politician Paul Painlevé is described. His contribution to the analytical theory of nonlinear differential equations was significant. The paper outlines the achievements of Paul Painlevé 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 Painlevé transcendents. The contribution of Paul Painlevé 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.

Keywords

mathematician politician Painlevé equations Painlevé transcendents Painlevé paradox 

MSC2010 numbers

01-00 01A55 01A60 

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References

  1. 1.
    Albouy, A., There Is a Projective Dynamics, Eur. Math. Soc. Newsl., 2013, no. 89, pp. 37–43.Google Scholar
  2. 2.
    Appell, P., Traité de Mécanique rationnelle: T. 2. Dynamique des systèmes. Mécanique analytique, 6th ed., Paris: Gauthier-Villars, 1953.Google Scholar
  3. 3.
    Borisov, A.V. and Mamaev, I. S., Generalized Problem of Two and Four Newtonian Centers, Celestial Mech. Dynam. Astronom., 2005, vol. 92, no. 4, pp. 371–380.CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Stability of Steady Rotations in the Non-Holonomic Routh Problem, Nelin. Dinam., 2006, vol. 2, no. 3, pp. 333–345 [Regul. Chaotic Dyn., 2008, vol. 13, no. 4, pp. 239–249].Google Scholar
  5. 5.
    Borisov, A.V. and Mamaev, I. S., Isomorphisms of Geodesic Flows on Quadrics, Nelin. Dinam., 2009, vol. 5, no. 2, pp. 145–158 [Regul. Chaotic Dyn., 2009, vol. 14, nos. 4–5, pp. 455–465].Google Scholar
  6. 6.
    Chaplygin, S.A., On the Theory ofMotion of Nonholonomic Systems. The Reducing-Multiplier Theorem, Mat. Sb., 1912, vol. 28, no. 2, pp. 303–314 [Regul. Chaotic Dyn., 2008, vol. 13, no. 4, pp. 369–376].Google Scholar
  7. 7.
    Chaplygin, S. A., On a Paraboloid Pendulum, in Complete Collection of Works: Vol. 1, Leningrad: Izd. Akad. Nauk SSSR, 1933, pp. 194–199 (Russian).Google Scholar
  8. 8.
    Conte, R. and Musette, M., The Painlevé Handbook, Dordrecht: Springer, 2008.zbMATHGoogle Scholar
  9. 9.
    de Sparre, Sur le frottement de glissement, C. R. Acad. Sci. Paris, 1905, vol. 141, pp. 310–312.Google Scholar
  10. 10.
    Diacu, F. and Holmes, Ph., Celestial Encounters: The Origins of Chaos and Stability, Princeton, NJ: Princeton Univ. Press, 1999.Google Scholar
  11. 11.
    Diacu, F., Singularities of the Newtonian N-Body Problem, in Classical and Celestial Mechanics: The Recife Lectures, H. Cabral, F. Diacu (Eds.), Princeton, NJ: Princeton Univ. Press, 2002.Google Scholar
  12. 12.
    Fuchs, R., Sur quelques équations différentielles linéaires du second ordre, C. R. Acad. Sci. Paris, 1905, vol. 141, pp. 555–558.Google Scholar
  13. 13.
    Fuchs, R., Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegene wesentlich singuläre Stellen, Math. Ann., 1907, vol. 63, no. 3, pp. 301–321.CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Gambier, B., Sur les équations différentielles de second ordre et du premier degré dont l’intégrale est à points critiques fixés, Acta Math., 1910, vol. 33, pp. 1–55.CrossRefMathSciNetGoogle Scholar
  15. 15.
    Gromak, V. I., Laine, I., and Shimomura, Sh., Painlevé Differential Equations in the Complex Plane, De Gruyter Stud. in Math., vol. 28, Berlin: de Gruyter, 2002.Google Scholar
  16. 16.
    Hamel, G., Bemerkungen zu den vorstehenden Aufsätzen der Herren F.Klein und R. v. Mises, Ztschr. f. Math. u. Physik, 1909, vol. 58, nos. 1–2, pp. 195–196.zbMATHGoogle Scholar
  17. 17.
    Jellett, J.H., A Treatise on the Theory of Friction, Dublin: Hodges, Foster, 1872.Google Scholar
  18. 18.
    Klein, F., Zu Painlevés Kritik der Coulombschen Reibgesetze, Ztschr. f. Math. u. Physik, 1910, vol. 58, nos. 1–2, pp. 186–191.Google Scholar
  19. 19.
    Kozlov, V.V., Nonexistence of Univalued Integrals and Branching of Solutions in Rigid Body Dynamics, Prikl. Mat. Mekh., 1978, vol. 42, no. 3, pp. 400–406 [J. Appl. Math. Mech., 1978, vol. 42, no. 3, pp. 420–426].Google Scholar
  20. 20.
    Kozlov, V.V., Tensor Invariants of Quasihomogeneous Systems of Differential Equations, and the Kovalevskaya - Lyapunov Asymptotic Method, Mat. Zametki, 1992, vol. 51, no. 2, pp. 46–52 [Math. Notes, 1992, vol. 51, no. 2, pp. 138–142].Google Scholar
  21. 21.
    Kozlov, V.V., Branching of the solutions and Polynomial Integrals of the Equations of Dynamics, Prikl. Mat. Mekh., 1998, vol. 62, no. 1, pp. 3–11 [J. Appl. Math. Mech., 1998, vol. 62, no. 1, pp. 1–8].zbMATHMathSciNetGoogle Scholar
  22. 22.
    Kouneiher, J., Géométrie au XXe siècle, 1930–2000: histoire et horizons, Paris: Hermann, 2005.Google Scholar
  23. 23.
    Kudryashov, N.A., The First and Second Painlevé Equations of Higher Order and Some Relations between Them, Phys. Lett. A, 1997, vol. 224, no. 6, pp. 353–360.CrossRefMathSciNetGoogle Scholar
  24. 24.
    Kudryashov, N.A. and Soukharev, M. B., Uniformization and Transcendence of Solutions for the First and Second Painlevé Hierarchies, Phys. Lett. A, 1998, vol. 237, nos. 4–5, pp. 206–216.CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Kudryashov, N.A., On New Transcendents Defined by Nonlinear Ordinary Differential Equations, J. Phys. A, 1998, vol. 31, no. 6, L129–L137.CrossRefMathSciNetGoogle Scholar
  26. 26.
    Kudryashov, N.A., Transcendents Defined by Nonlinear Fourth-Order Ordinary Differential Equations, J. Phys. A, 1999, vol. 32, no. 6, pp. 999–1013.CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Kudryashov, N.A., Two Hierarchies of Ordinary Differential Equations and Their Properties, Phys. Lett. A, 1999, vol. 252, nos. 3–4, pp. 173–179.CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Kudryashov, N.A., Amalgamations of the Painlevé Equations, J. Math. Phys., 2003, vol. 44, no. 12, pp. 6160–6178.CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Kudryashov, N.A., Analytical Theory of Nonlinear Differential Equations, Moscow-Izhevsk: Institute of Computer Science, 2004 (Russian).Google Scholar
  30. 30.
    Kudryashov, N.A., Methods of Nonlinear Mathematical Physics, Moscow: Intellekt, 2010 (Russian).Google Scholar
  31. 31.
    Kudryashov, N.A., Higher Painlevé Transcendents As Solutions of Some Nonlinear Integrable Hierarchies, Regul. Chaotic Dyn., 2014, vol. 19, no. 1, pp. 48–63.CrossRefMathSciNetGoogle Scholar
  32. 32.
    Lecornu, L., Sur le frottement de glissement, C. R. Acad. Sci. Paris, 1905, vol. 140, pp. 635–637.zbMATHGoogle Scholar
  33. 33.
    Lemaître, G., L’univers en expansion, Ann. Soc. Sci. Bruxelles A, 1933, vol. 53, pp. 51–85.Google Scholar
  34. 34.
    Lyapunov, A. M., On a Certain Property of the Differential Equations of Motion of a Heavy Rigid Body with a Fixed Point, Soobshch. Kharkov. Mat. Obshch., Ser. 2, 1894, vol. 4, pp. 123–140 (Russian).Google Scholar
  35. 35.
    Mandel, J., Cours de mécanique des milieux continus: T. 1. Généralités. Mécanique des fluides, Paris: Gauthier-Villars, 1966.Google Scholar
  36. 36.
    Mather, J. and McGehee, R., Solutions of the Collinear Four-Body Problem Which Become Unbounded in Finite Time, in Dynamical Systems Theory and Applications, J. Moser (Ed.), Lect. Notes Phys. Monogr., vol. 38, Berlin: Springer, 1975, pp. 573–587.CrossRefGoogle Scholar
  37. 37.
    Mises, R., Zur Kritik der Reibungsgesetze, Ztschr. f. Math. u. Physik, 1909, vol. 58, nos. 1–2, pp. 191–194.zbMATHGoogle Scholar
  38. 38.
    Morales-Ruiz, J. J., Kovalevskaya, Liapounov, Painlevé, Ziglin and the Differential Galois Theory, Regul. Chaotic Dyn., 2000, vol. 5, no. 3, pp. 251–272.CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Morales-Ruiz, J. J. and Ramis, J.-P., Integrability of Dynamical Systems through Differential Galois Theory: A Practical Guide, in Differential Algebra, Complex Analysis and Orthogonal Polynomials, P. B. Acosta-Humánez, F. Marcellán (Eds.), Contemp. Math., vol. 509, Providence, RI: AMS, 2010, pp. 143–220.CrossRefGoogle Scholar
  40. 40.
    O’Connor, J. J. and Robertson, E.F., Paul Painlevé, http://www-history.mcs.st-and.ac.uk/Biographies/Painleve.html
  41. 41.
    Painlevé, P., Sur les lignes singuliéres des fonctions analytiques, Doctoral dissertation (Thése 1), Paris: Gauthier-Villars, 1887.Google Scholar
  42. 42.
    Painlevé, P., Sur la transformation des fonctions harmoniques et les systèmes triples de surfaces orthogonales, Travaux et mémoires de la Faculté de sciences de Lille, 1889, vol. 1, pp. 1–29.Google Scholar
  43. 43.
    Painlevé, P., Mémoire sur les équations différentielles du premier ordre, Ann. sci. de l’ É.N. S. Sér. 3, 1891, vol. 8, pp. 9–58, 103–140, 201–226, 267–284; 1892, vol. 9, pp. 9–30, 101–144, 283–308.zbMATHGoogle Scholar
  44. 44.
    Painlevé, P., Sur la transformation des équations de la Dynamique, C. R. Acad. Sci. Paris, 1892, vol. 115, pp. 714–717, 874–875.Google Scholar
  45. 45.
    Painlevé, P., Mémoire sur la transformation des équations de la Dynamique, J. Math. Pures Appl. (4), 1894, vol. 10, pp. 5–92.zbMATHGoogle Scholar
  46. 46.
    Painlevé, P., Sur les mouvements et les trajectoires réels des systèmes, Bull. Soc. Math. France, 1894, vol. 22, pp. 136–184.zbMATHMathSciNetGoogle Scholar
  47. 47.
    Painlevé, P., Leçons sur l’intégration des équations différentielles de la mécanique et applications, Paris: Hermann, 1895.zbMATHGoogle Scholar
  48. 48.
    Painlevé, P., Leçons sur le frottement, Paris: Hermann, 1895.zbMATHGoogle Scholar
  49. 49.
    Painlevé, P., Leçons sur la théorie analytique des équations différentielles, Paris: Hermann, 1897.Google Scholar
  50. 50.
    Painlevé, P., Mémoire sur les intégrales premières du ploblème des n corps, Bull. Astronomique, Sér. 1, 1898, vol. 15, pp. 81–113.Google Scholar
  51. 51.
    Painlevé, P., Mémoire sur les équations différentielles dont l’intégrale générale est uniforme, Bull. Soc. Math. Phys. France, 1900, vol. 28, pp. 201–261.zbMATHGoogle Scholar
  52. 52.
    Painlevé, P., Sur les équations différentielles du second ordre et d’ordre supérieur dont l’intégrale générale est uniforme, Acta Math., 1900, vol. 23, pp. 1–80; 1902, vol. 25, pp. 1–85.CrossRefMathSciNetGoogle Scholar
  53. 53.
    Painlevé, P., Sur le développement des fonctions analytiques, in Borel, É., Leçons sur les fonctions de variables réelles et les développements en série de polynômes, Paris: Gauthier-Villars, 1905, pp. 101–147.Google Scholar
  54. 54.
    Painlevé, P., Sur les lois de frottement de glissement, C. R. Acad. Sci. Paris, 1905, vol. 140, pp. 702–707; vol. 141, pp. 401–405; vol. 141, pp. 546–552.zbMATHGoogle Scholar
  55. 55.
    Painlevé, P., Sur les équations différentielles du second ordre à point critiques fixes, C. R. Acad. Sci. Paris, 1906, vol. 143, pp. 1111–1117.Google Scholar
  56. 56.
    Painlevé, P., Sur les équations différentielles du premier ordre dont l’intégrale générale n’a qu’un nombre fini de branches, in Boutroux, P., Leçons sur les fonctions définies par les équations différentielles du premier ordre, Paris: Gauthier-Villars, 1908, pp. 141–187.Google Scholar
  57. 57.
    Painlevé, P., Les axiomes de la mécanique, examen critique, avec une note sur la propagation de la lumière, Paris: Gauthier-Villars, 1922. (Réimpiré par les Éditions Jacques Gabay, Paris, 1995.)Google Scholar
  58. 58.
    Pérès, J., Mécanique générale, 2nd ed., Paris: Masson, 1962.Google Scholar
  59. 59.
    Pfeifer, F., Zur Frage der sogenannten Coulombschen Reibungsgesetze, Ztschr. f. Math. u. Physik, 1909, vol. 58, no. 3, pp. 273–311.Google Scholar
  60. 60.
    Picard, E., Mémoire sur la théorie des fonctions algébriques de deux variables, J. Math. Pures Appl. (4), 1889, vol. 5, pp. 135–320.Google Scholar
  61. 61.
    Poincaré, H., La science et l’hypothèse, Paris: Flammarion, 1906; Poincaré, H., Science et méthode, Paris: Flammarion, 1908; Poincaré, H., La valeur de la science, Paris: Flammarion, 1912; Poincaré, H., Dernières pensées, Paris: Flammarion, 1913; Poincaré, H., La mécanique nouvelle, livre réunissnat en un seul volume le texte d’une conférence faite au congrès de Lille de l’Association française pour l’avancement des sciencex en 1909, le mémoire du 23 juillet 1905 intituté Sur la dynamique de l’électron, publié aux Rendiconti del Circolo matematico di Palermo XXI (1906) et une Note aux Comptes Rendus de l’Académie des Sciences, de même titre (séance du 15 juin 1905, CXL, 1905, p. 1504); Paris: Gauthier-Villars, 1924; réimprimé par les Éditions Jaques Gabay, Paris, 1989; Poincaré, H., Les méthodes nouvelles de la mécanique céleste: Tomes I, II et III, Paris: Gauthier-Villars, 1892, 1893, 1899; réimprimé par la Librairie scientifique et technique Albert Blanchard, Paris, 1987.Google Scholar
  62. 62.
    Prandtl, L., Bemerkungen zu den Aufsätzen der Herren F.Klein, R. v.Mises und G. Hamel. Ztschr. f. Math. u. Physik, 1909, vol. 58, nos. 1–2, pp. 196–197.zbMATHGoogle Scholar
  63. 63.
    Saari, D.G., Collisions, Rings, and Other Newtonian N-Body Problems, CBMS Region. Conf. Ser. Math., vol. 104, Providence,RI: AMS, 2005.zbMATHGoogle Scholar
  64. 64.
    Stewart, D. E., Dynamics with Inequalities: Impacts and Hard Constraints, Philadelphia, PA: SIAM, 2011.CrossRefGoogle Scholar
  65. 65.
    Topalov, P. I. and Matveev, V. S., Geodesical Equivalence and the Liouville Integration of the Geodesic Flows, Regul. Chaotic Dyn., 1998, vol. 3, no. 2, pp. 30–45.CrossRefzbMATHMathSciNetGoogle Scholar
  66. 66.
    Wintner, A., The Analytical Foundations of Celestial Mechanics, Princeton, NJ: Princeton Univ. Press, 1952.Google Scholar
  67. 67.
    Xia, Zh., The Existence of Noncollision Singularities in Newtonian Systems, Ann. of Math. (2), 1992, vol. 135, no. 3, pp. 411–468.CrossRefzbMATHMathSciNetGoogle Scholar
  68. 68.
    Yoshida, H., Necessary Condition for the Existence of Algebraic First Integrals: 1. Kowalevski’s Exponents, Celestial Mech., 1983, vol. 31, no. 4, pp. 363–379.CrossRefzbMATHMathSciNetGoogle Scholar
  69. 69.
    Ziglin, S. L., Branching of Solutions and Nonexistence of First Integrals in Hamiltonian Mechanics: 1, Funkts. Anal. Prilozh., 1982, vol. 16, no. 3, pp. 30–41 [Funct. Anal. Appl., 1982, vol. 16, no. 3, pp. 181–189].MathSciNetGoogle Scholar

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© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Institute of Computer ScienceUdmurt State UniversityIzhevskRussia
  2. 2.A.A. Blagonravov Mechanical Engineering Research Institute of RASMoscowRussia
  3. 3.Institute of Mathematics and Mechanics of the Ural Branch of RASEkaterinburgRussia
  4. 4.Department of Applied MathematicsNational Research Nuclear University “MEPhI”MoscowRussia

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