TITLE:
New families of Solutions in $N$--Body Problems

AUTHOR:
Carles Sim\'o
Departament de Matem\`atica Aplicada i An\`alisi, Universitat de Barcelona,
Gran Via, 585, 08007 Barcelona, Spain
e-mail: carles@maia.ub.es

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 3 bodies.
Simple solutions, as the so called {\em 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 {\em 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.
