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.