n-body choreography
An n-body choreography is a periodic solution to the n-body problem in which all the bodies are equally spread out along a single orbit.[1] The term was originated in 2000 by Chenciner and Montgomery.[1][2][3] One such orbit is a circular orbit, with equal masses at the corners of an equilateral triangle; another is the figure-8 orbit, first discovered numerically in 1993 by Cristopher Moore[4] and subsequently proved to exist by Chenciner and Montgomery. Choreographies can be discovered using variational methods,[1] and more recently, topological approaches have been used to attempt a classification in the planar case.[5]
References
- ^ a b c Vanderbei, Robert J. (2004). "New Orbits for the n-Body Problem". Annals of the New York Academy of Sciences. 1017 (1): 422–433. arXiv:astro-ph/0303153. Bibcode:2004NYASA1017..422V. CiteSeerX 10.1.1.140.6108. doi:10.1196/annals.1311.024. PMID 15220160. S2CID 8202325.
- ^ Simó, C. [2000], New families of Solutions in N-Body Problems, Proceedings of the ECM 2000, Barcelona (July, 10-14).
- ^ "A remarkable periodic solution of the three-body problem in the case of equal masses". The original article by Alain Chenciner and Richard Montgomery. Annals of Mathematics, 152 (2000), 881–901.
- ^ Moore, Cristopher (1993-06-14). "Braids in classical dynamics". Physical Review Letters. 70 (24). American Physical Society (APS): 3675–3679. Bibcode:1993PhRvL..70.3675M. doi:10.1103/physrevlett.70.3675. ISSN 0031-9007. PMID 10053934. Moore's numerical discovery of the figure-8 choreography using variational methods.
- ^ Montaldi, James; Steckles, Katerina (2013). "Classification of symmetry groups for planar n-body choreographies". Forum of Mathematics, Sigma. 1. Cambridge University Press (CUP): e5. arXiv:1305.0470. Bibcode:2013arXiv1305.0470M. doi:10.1017/fms.2013.5. ISSN 2050-5094.
External links
- Cris Moore's 1993 paper
- Some animations of 2-d and 3-d orbits, including the figure-8
- Greg Minton's choreography applet which allows the user to search numerically for orbits of their own design
- Animation of the solution with three bodies following each other in a figure of eight orbit
- Youtube videos of several n-body choreographies
- Feature column on Simó's Choreographies for the American Mathematical Society
- A collection of animations of planar choreographies
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