European Congress of Mathematics pp 101-115 | Cite as

# New Families of Solutions in *N*-Body Problems

## Abstract

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.

## Keywords

Periodic Solution Double Point Small Loop Newtonian Potential Central Configuration## Preview

Unable to display preview. Download preview PDF.

## References

- [1]G. D. Birkhoff,
*Dynamical Systems*, Amer. Math. Soc., 1927.zbMATHGoogle Scholar - [2]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 - [3]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 - [4]A. Chenciner, R. Montgomery,
*A remarkable periodic solution of the three body problem in the case of equal masses*, Annals of Mathematics,**152**(2000), 881–901.MathSciNetzbMATHCrossRefGoogle Scholar - [5]J. Gerver, private communication, (2000).Google Scholar
- [6]M. Hénon, private communication, (2000).Google Scholar
- [7]R. Moeckel,
*On central configurations*, Math. Zeit.,**205**(1990), 499–517.MathSciNetzbMATHCrossRefGoogle Scholar - [8]C. Simó,
*Relative equilibrium solutions in the four-body problem*, Cel. Mechanics,**18**(1978), 165–184.zbMATHCrossRefGoogle Scholar - [9]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 - [10]J. Stoer, R. Bulirsch,
*Introduction to Numerical Analysis*, Springer-Verlag, 1983 (second printing).Google Scholar