New Perspectives on the N -body Problem
The problem of the motion of N ≥ 2 point–masses (i.e., ideal bodies with no physical dimension identified with points in the Euclidean three–dimensional space) interacting only through Newton’s law of mutual gravitational attraction, has been a central issue in astronomy, physics and mathematics since the early developments of modern calculus. When N = 2 the problem has been completely solved (“integrated”) by Newton: the motion takes place on conics, whose focus is occupied by the center of mass of the two bodies; but for N ≥ 3 a complete understanding of the problem is still far away, notwithstanding the efforts of many excellent mathematicians over the last centuries (Newton, Euler, d’Alembert, Lagrange, Laplace followed by H. Poincare, G.D. Birkhoff, C.L. Siegel, J.K. Moser, V.I. Arnold, M. Herman,...) The dynamics f the N–body problem has the fascinating feature of hosting regular and chaotic regimes and has often inspired the elaboration of new mathematical theories of local, perturbative and global nature. Indeed, to be attacked efficiently, it requires a sophisticated mixture of algebraic, analytical, geometrical, topological and probabilistic techniques. While the original impulse, coming from astronomy, has been somehow shaded by the massive use of machines for computing orbits of celestial bodies or satellites, the mathematical richness and beauty of the N–body problem has retained most of its original attraction, and still may be considered as one of the most fascinating and difficult branch of mathematics (see, below, for a list of recent excellent contributions, most of the authors being present at this conference).