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Three body problem in early 20th century  

Lee, Ho Joong (Department of Basic Science, Hong Ik University)
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Journal for History of Mathematics / v.25, no.4, 2012 , pp. 53-67 More about this Journal
Today, it is necessary to calculate orbits with high accuracy in space flight. The key words of Poincar$\acute{e}$ in celestial mechanics are periodic solutions, invariant integrals, asymptotic solutions, characteristic exponents and the non existence of new single-valued integrals. Poincar$\acute{e}$ define an invariant integral of the system as the form which maintains a constant value at all time $t$, where the integration is taken over the arc of a curve and $Y_i$ are some functions of $x$, and extend 2 dimension and 3 dimension. Eigenvalues are classified as the form of trajectories, as corresponding to nodes, foci, saddle points and center. In periodic solutions, the stability of periodic solutions is dependent on the properties of their characteristic exponents. Poincar$\acute{e}$ called bifurcation that is the possibility of existence of chaotic orbit in planetary motion. Existence of near exceptional trajectories as Hadamard's accounts, says that there are probabilistic orbits. In this context we study the eigenvalue problem in early 20th century in three body problem by analyzing the works of Darwin, Bruns, Gyld$\acute{e}$n, Sundman, Hill, Lyapunov, Birkhoff, Painlev$\acute{e}$ and Hadamard.
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1 H. Poincare,"Sur le probleme des trois corps et les equations de la dynamique", Acta Mathematica 13, (1890), pp. 1-270; OEuvres VII, pp. 262-479.
2 Gautier, A., Essai historique sur le probleme des trois corps, Paris, Veuve Courcier, pp. 79-80, 1817.
3 R. Taton and C. Wilson, The General History of Astronomy, (Cambridge University press, 1989)Vol. II.
4 C Wilson,"The three-body problem"; Companion Encyclopedia of the Theory and Philosophy of the Mathematical Sciences(Routledge, Newyork, 1994)vol 2, pp. 1059-1061.
7 June Barrow Green, Poincare and the Three Body Problem, American Mathematical Society, Providence, 1997.
8 E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Reprinted, Cambridge University Press, 1970.
9 일리야 프리고진, 이자벨 스텐저스 (신국조 옮김), 혼돈으로부터의 질서, 고려원미디어, p. 360, 1993.
10 E. T. Bell, Men of Mathematics, Victor Gollancz, London, pp. 599-600, 1939.
11 Henri Poincare, Henri Poincare, Les methodes nouvelles de la Mecanique Celeste; 영역 본, New Methods of Celestial Mechanics, Edited and Introduced by Daniel L. Goroff, I 80.
12 Victor G. Szebehely, Hans Mark; Adventure in Celestial Mechanics, John wiley & son, 1998, 1장과 13장
13 P. S. Laplace, Mecanique celeste, vol I-V.
14 Jacques Hadamard,"L'oeuvre mathematique de Poincare; The Mathematical Heritage of Henri Poincare", Proceedings of Symposia in Pure Mathematics(American Mathematical Society, USA, 1983) volume 39 part 2.
15 Hans-Heinrich Voigt, Outline of Astronomy I, II; 유경로외 5인 공역, 천문학강요, 일신사, pp. 50-51, 1992.
16 Mauri Valtonen and Hannu Karttunen, The Three-Body Problem, Cambridge University Press, 2006.
17 최규홍, 천체역학, 민음사, 1997.
18 Archie E. Roy and Bonnie A Steves, From Newton to Chaos(Plenum Press, New york, 1995).
19 C.Wilson, "The dynamic of Solar System", Companion Encyclopedia of the Theory and Philosophy of the Mathematical Sciences(Routledge, Newyork, 1994)vol2, pp. 1044-1053.
20 Barrow-Green,"The dramatic episode of Sundman", Historia Mathematica, 37(2), pp. 164-203, 2010.   DOI   ScienceOn
21 Barrow-Green, June (2005). Henri Poincare, memoir on the three-body problem, In: Grattan-Guinness, I. ed. Landmark Writings in Western Mathematics 1640-1940. Amsterdam: Elsevier, pp. 627-638, 1890.