Poincare and the Three Body Problem

The purpose of the thesis is to present an account of Henri Poincare's famous memoir on the three body problem, the final version of which was published in Acta Mathematica in 1890 as the prize-winning entry in King Oscar II's 60th birthday competition. The memoir is reknowned both for its role in providing the foundations for Poincare's celebrated three volume Methodes Nouvelles de la Mecanique Celeste, and for containing the first mathematical description of chaotic behaviour in a dynamical system. A historical context is provided both through consideration of the problem itself and through a discussion of Poincare's earlier work which relates to the mathematics developed in the memoir. The organisation of the Oscar competition, which was undertaken by Gosta Mittag-Leffler, is also described. This not only provides an insight into the late 19th century European mathematical community but also reveals that after the prize had been awarded Poincare found an important error in his work and substantially revised the memoir prior to its publication in Acta. The discovery of a printed version of the original memoir personally annotated by Poincare has allowed for a detailed comparative study of the mathematics contained in both versions of the memoir. The error is explained and it is shown that it was only as a result of its correction that Poincare discovered the chaotic behaviour now associated with the memoir. The contemporary reception of the memoir is discussed and Poincare's subsequent work in celestial mechanics and related topics is examined. Through the consideration of sources up to 1920 the influence and impact of the memoir on the progress of the three body problem and on dynamics in general is assessed.

[1]  R. J. Adcock.  On the Stability of the Solar System , 1874 .

[2]  Demonstration of the Differential Equations Employed by Delaunay in the Lunar Theory , 1876 .

[3]  Sur la détermination des distances mutuelles dans le problème des trois corps , 1884 .

[4]  Sur un moyen d'augmenter la convergence des séries trigonométriques , 1886 .

[5]  On the convergence of the series used in the subject of perturbations , 1896 .

[6]  REMARKS ON THE PROGRESS OF CELESTIAL MECHANICS SINCE THE MIDDLE OF THE CENTURY. , 1896, Science.

[7]  C. V. L. Charlier,et al.  Periodic Orbits , 1898, Nature.

[8]  On the Solution of Dynamical Problems in terms of Trignometric Series , 1901 .

[9]  On certain Families of Periodic Orbits , 1909 .

[10]  On Poincaré’s correction to Bruns’ theorem , 1913 .

[11]  The rôle of the point-set theory in geometry and dynamics , 1915 .

[12]  George D. Birkhoff,et al.  The restricted problem of three bodies , 1915 .

[13]  G D Birkhoff,et al.  Dynamical Systems with Two Degrees of Freedom. , 1917, Proceedings of the National Academy of Sciences of the United States of America.

[14]  Harold Marston Morse.  A One-to-One Representation of Geodesics on a Surface of Negative Curvature , 1921 .

[15]  George D. Birkhoff,et al.  Surface transformations and their dynamical applications , 1922 .

[16]  On Poincaré's Theorem of “The non-existence of Uniform Integrals of Dynamical Equations” , 1924 .

[17]  George D. Birkhoff,et al.  On the periodic motions of dynamical systems , 1927, Hamiltonian Dynamical Systems.

[18]  On Periodic Solutions of Hamiltonian Systems of Differential Equations , 1928 .

[19]  M. Morse.  The Calculus of Variations in the Large , 1934 .

[20]  Marston Morse,et al.  George David Birkhoff and his mathematical work , 1946 .

[21]  On periodic orbits in the restricted problem of three bodies , 1950 .

[22]  Hans Hornich,et al.  Zur Theorie der partiellen Differentialgleichungen , 1951 .

[23]  L. Cesari,et al.  Asymptotic Behaviour and Stability Problems in Ordinary Differential Equations , 1960 .

[24]  V. Arnold.  SMALL DENOMINATORS AND PROBLEMS OF STABILITY OF MOTION IN CLASSICAL AND CELESTIAL MECHANICS , 1963 .

[25]  Richard McGehee,et al.  Solutions of the collinear four body problem which become unbounded in finite time , 1975 .

[26]  A. Schlissel.  The development of asymptotic solutions of linear ordinary differential equations, 1817–1920 , 1977, Archive for History of Exact Sciences.

[27]  Raoul Bott,et al.  Marston Morse and his mathematical works , 1980 .

[28]  Mary Tiles,et al.  Georg Cantor: His Mathematics and Philosophy of the Infinite. , 1982 .

[29]  J. J. Gray,et al.  Fuchs and the theory of differential equations , 1984 .

[30]  Donald G. Saari.  A visit to the newtonian N-body problem via elementary complex variables , 1990 .

[31]  P. Holmes.  Poincaré, celestial mechanics, dynamical-systems theory and “chaos” , 1990 .

[32]  Joseph L. Gerver.  The existence of pseudocollisions in the plane , 1991 .

[33]  Jacobi and the birth of Lie's theory of groups , 1991 .

[34]  Zhihong Xia,et al.  The existence of noncollision singularities in newtonian systems , 1992 .

[35]  Ueber die Druckkräfte, welche auf Ringe wirksam sind, die in bewegte Flüssigkeit tauchen. , .