European Congress of Mathematics pp 101-115

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

DOI: 10.1007/978-3-0348-8268-2_6

- Cite this paper as:
- Simó C. (2001) New Families of Solutions in
*N*-Body Problems. In: Casacuberta C., Miró-Roig R.M., Verdera J., Xambó-Descamps S. (eds) European Congress of Mathematics. Progress in Mathematics, vol 201. Birkhäuser, Basel

## 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.

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