Abstract
A beautiful plane eight-shaped orbit has been found by Alain Chenciner, Richard Montgomery and Caries Simo through the minimisation of the action between suitable limit conditions. The three masses are equal and chase each other along the eight shape. This procedure can be generalized and leads to a family of three-dimensional periodic orbits with three equal masses and with 12 space-time symmetries per period. The property of a unique orbit for the three masses is conserved in a suitable uniformly rotating set of axes. The eight-shaped orbit represents the end of the family, its beginning being the classical Lagrangian solution with three equal masses and with a uniformly rotating equilateral triangle.
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References
Alain Chenciner and Richard Montgomery: 1999, ‘A remarkable periodic solution of the three-body problem in the case of equal masses’, Northwestern University, Evanston conference on Celestial Mechanics, December 15–19.
Pontryagin, L. S., Boltyansky, V. G., Gamkrelidze, R. V. and Mischenko, E. F.: 1962, The Mathematical Theory of Optimal Processes, Chap. 1–5, Interscience Publishers, (division of Wiley) New York, London.
Marchai, C.: 1990, The Three-body Problem,Elsevier Science Publishers B.V., pp. 302–311.
Henri Poincaré: 1952, Oeuvres de Henri Poincaré, Tome 7, Gauthier-Villars Editeur, Paris, pp. 253–254. Also Bulletin Astronomique, Tome 1, Février, 1884, 65–66.
Marchai, C.: 1990, The Three-body Problem,Elsevier Science Publishers B.V., pp 251–257.
Marchal, C.: 1990, The Three-body Problem, Elsevier Science Publishers B.V., pp. 25–29.
Marchai, C.: 1975, Survey paper. Second-order tests in optimization theories. J Optim Theory Appli(JOTA) 15 (6), 633–666
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© 2001 Springer Science+Business Media Dordrecht
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Marchal, C. (2001). The Family P 12 of the Three-Body Problem — the Simplest Family of Periodic Orbits, with Twelve Symmetries per Period. In: Dvorak, R., Henrard, J. (eds) New Developments in the Dynamics of Planetary Systems. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2414-2_18
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DOI: https://doi.org/10.1007/978-94-017-2414-2_18
Publisher Name: Springer, Dordrecht
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