New Families of Solutions in N-Body Problems
The N-body problem is one of the outstanding classical problems in Mechanics and other sciences. In the Newtonian case few results are known for the 3-body problem and they are very rare for more than three bodies. Simple solutions, such as the so-called relative equilibrium solutions, in which all the bodies rotate around the center of mass keeping the mutual distances constant, are in themselves a major problem. Recently, the first example of a new class of solutions has been discovered by A. Chenciner and R. Montgomery. Three bodies of equal mass move periodically on the plane along the same curve. This work presents a generalization of this result to the case of N bodies. Different curves, to be denoted as simple choreographies, have been found by a combination of different numerical methods. Some of them are given here, grouped in several families. The proofs of existence of these solutions and the classification turn out to be a delicate problem for the Newtonian potential, but an easier one in strong force potentials.
KeywordsPeriodic Solution Double Point Small Loop Newtonian Potential Central Configuration
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- A. Chenciner, N. Desolneux, Minima de l’intégrale d’action et équilibres relatifs de n corps, C.R.A.S. Paris, 326, Série I (1998), 1209–1212. Correction in C.R.A.S. Paris, 327, Série I (1998), 193.Google Scholar
- A. Chenciner, J. Gerver, R. Montgomery, C. Simó, Simple choreographic motions of N bodies with strong forces, to appear in Geometry, Mechanics, and Dynamics, Springer-Verlag.Google Scholar
- J. Gerver, private communication, (2000).Google Scholar
- M. Hénon, private communication, (2000).Google Scholar
- C. Simó, Analytical and numerical computation of invariant manifolds. In D. Benest et C. Froeschlé, editors, Modern methods in celestial mechanics, 285–330, Editions Frontières, 1990.Google Scholar
- J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, Springer-Verlag, 1983 (second printing).Google Scholar