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A Walk Through the New Methods of Celestial Mechanics

  • Alain ChencinerEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 54)

Abstract

En ce qui concerne le problème des trois corps, je ne suis pas sorti du cas suivant: Je considère trois masses, la première très grande, la seconde petite mais finie, la troisième infiniment petite; je suppose que les deux premières décrivent un cercle autour de leur centre de gravité commun et que la troisième se meut dans le plan de ces cercles. Tel serait le cas d’une petite planète troublée par Jupiter, si l’on négligeait l’excentricité de Jupiter et l’inclinaison des orbites. (Concerning the Three-Body Problem, I did not venture away from the following case: I consider three masses, the first one very big, the second one small but finite, the third one infinitesimal; I suppose that the first two describe a circle around their common center of mass and that the third moves in the plane of these circles. Such would be the case of a small planet perturbed (maybe “troubled” better conveys the charm of the eighteen century flavoured “troublé”) by Jupiter if one was neglecting Jupiter’s eccentricity and the inclination of the orbits.)

This is how, at the beginning of his memoir On the three-body problem and the equations of dynamics which, in 1889, wins the prize given by the king Oscar of Sweden, Poincaré describes the setting which will be developped in The New Methods of Celestial Mechanics. This work was indeed built in order to correct the sentence which, in the first version of the memoir, followed the one above:

Dans ce cas particulier, j’ai démontré rigoureusement la stabilité. (In this particular case, I have rigorously proved stability.)

Respectively published by Gauthier-Villars in 1892, 1893 et 1899 these three volumes (1,268 pages + 10 pages for the tables of contents) will cause Paul Appell to say in 1925: “It is likely that, during the next half-century, this book will be the mine from which more modest researchers will extract their material”. The prediction came true: more than a century later, we contemplate some of these nuggets, whose brightness has not weakened. (This text is essentially a translation by the author of A. Chenciner [7] (Une promenade dans les Méthodes nouvelles de la mécanique céleste, Gazette des Mathématiciens, n 0134, October 2012, 37–47, and Quadrature, special issue dedicated to Henri Poincaré, November 2012). A much more technical text on the same subject is A. Chenciner (Poincaré and the Three-Body Problem, “Poincaré seminar” (also called “Bourbaphy”), XVI, November 2012, 45–133, www.bourbaphy.fr/novembre2012.html). As in this last text, I have chosen to leave in French Poincaré’s quotations, the translation being given in footnotes. The hope is that some readers be given the chance to appreciate Poincaré’s superb style.)

Keywords

Periodic Solution Kepler Problem Integral Invariant Lunar Theory Lindstedt Series 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Observatoire de Paris, IMCCE (UMR 8028), ASDParisFrance
  2. 2.Département de mathématiqueUniversité Paris VIIParisFrance

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