Abstract
A “simple choreography” for an N-body problem is a periodic solution in which all N masses trace the same curve without colliding. We shall require all masses to be equal and the phase shift between consecutive bodies to be constant. The first 3-body choreography for the Newtonian potential, after Lagrange’s equilateral solution, was proved to exist by Chenciner and Montgomery in December 1999 (Chenciner and Montgomery [2000]). In this paper we prove the existence of planar N-body simple choreographies with arbitrary complexity and/or symmetry, and any number N of masses, provided the potential is of strong force type (behaving like 1/ra, a≥2 as r→0). The existence of simple choreographies for the Newtonian potential is harder to prove, and we fall short of this goal. Instead, we present the results of a numerical study of the simple Newtonian choreographies, and of the evolution with respect to a of some simple choreographies generated by the potentials 1/ra, focusing on the fate of some simple choreographies guaranteed to exist for a≥2 which disappear as a tends to 1.
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References
Chenciner, A. [2001], Are there perverse choreographies?, to appear in the Proceedings of the HAMSYS Conference, Guanajuato (March 19–23, 2001), World Scientific Publishing Co., Singapore.
Chenciner, A. [2002], Action minimizing periodic orbits in the Newtonian n-body problem, in Celestial Mechanics, dedicated to Donald Saari for his 60th birthday; (A. Chenciner, R. Cushman, C. Robinson and J. Xia, eds.) Contemporary Mathematics 292, Amer. Math. Soc., 71–90.
Chenciner, A. and Montgomery, R. [2000], A remarkable periodic solution of the three body problem in the case of equal masses, Annals of Mathematics 152, 881–901.
Chenciner, A. and Venturelli, A. [2000], Minima de l’intégrale d’action du Problème newtonien de 4 corps de masses égales dans ℝ3: orbites “hip-hop”, Celestial Mechanics 77, 139–152.
Davies, I., Truman, A. and Williams, D. [1983], Classical periodic solutions of the equal mass 2N-body Problem, 2n-Ion problem, and the n-electron atom problem, Physics Letters 99A, 15–17.
Heggie, D. C. [2000], A new outcome of binary-binary scattering, Mon. Not. R. Astron. Soc. 318, L61–L63.
Hénon, M. [1976], A family of periodic solutions of the planar three-body problem, and their stability, Celestial Mechanics 13, 267–285.
Hénon, M. [2000], private communication.
Hoynant, G. [1999], Des orbites en forme de rosette aux orbites en forme de pelote, Sciences 99, 3–8.
Lagrange, J. [1772], Essai sur le problème des trois corps, Œuvres, Vol. 6, p. 273.
Montgomery, R. [1998], The N-body problem, the braid group, and action-minimizing periodic solutions, Nonlinearity 11, 363–376.
Montgomery, R. [2002], Action spectrum and Collisions in the three-body problem, in Celestial Mechanics, dedicated to Donald Saari for his 60th birthday; (A. Chenciner, R. Cushman, C. Robinson and J. Xia, eds.) Contemporary Mathematics 292, Amer. Math. Soc., 173–184.
Moore, C. [1993], Braids in Classical Gravity, Physical Review Letters 70, 3675–3679.
Palais, R. [1979], The principle of symmetric criticality, Comm. Math. Phys. 69, 19–30.
Poincaré, H. [1896], Sur les solutions périodiques et le principe de moindre action, C.R.A.S. Paris 123, 915–918 (30 Novembre 1896).
Simó, C. [2001a], New families of Solutions in N-Body Problems, Proceedings of the Third European Congress of Mathematics, (C. Casacuberta et al., eds.) Progress in Mathematics 201, 101–115, Birkäuser, Basel.
Simó, C. [2001b], Periodic orbits of the planar N-body problem with equal masses and all bodies on the same path, in The Restless Universe, (B. Steves and A. Maciejewski, eds.), 265–284, Institute of Physics Publ., Bristol.
Simó, C. [2002], Dynamical properties of the figure eight solution of the three-body problem, in Celestial Mechanics, dedicated to Donald Saari for his 60th birthday; (A. Chenciner, R. Cushman, C. Robinson and J. Xia, eds.) Contemporary Mathematics 292, Amer. Math. Soc., 209–228.
Stewart, I. [1996], Symmetry Methods in Collisionless Many-Body Problems, J. Nonlinear Sci. 0, 543–563.
Stoer, J. and Bulirsch, R. [1983], Introduction to Numerical Analysis, Springer-Verlag, 1983 (second printing).
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To Jerry Marsden on the occasion of his 60th birthday
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© 2002 Springer-Verlag New York, Inc.
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Chenciner, A., Gerver, J., Montgomery, R., Simó, C. (2002). Simple Choreographic Motions of N Bodies: A Preliminary Study. In: Newton, P., Holmes, P., Weinstein, A. (eds) Geometry, Mechanics, and Dynamics. Springer, New York, NY. https://doi.org/10.1007/0-387-21791-6_9
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DOI: https://doi.org/10.1007/0-387-21791-6_9
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