The three- and the N-body problem

Jacques Fejoz (1, 2, 3)

1 IMCCE - Institut de Mécanique Céleste et de Calcul des Ephémérides
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 ASD - Astronomie et systèmes dynamiques

We introduce the $N$-body problem of mathematical celestial mechanics, and discuss its astronomical relevance, its simplest solutions inherited from the two-body problem (called homographic motions and, among them, homothetic motions and relative equilibria), Poincaré's classification of periodic solutions, symmetric solutions and in particular choreographies such as the figure-eight solution, some properties of the global evolution and final motions, Chazy's classification in the three-body problem, some non-integrability results, perturbations series of the planetary problem and a short account on the question of its stability.

Mots clés: Newton's equation; symmetry; reduction; Conley-Wintner endomorphism; stability; planetary problem; Hill's problem; central configuration; homographic motions; relative equilibria; homothetic motion; periodic orbit; Poincaré's classification; choreography; figure-eight solution; Lagrangian action; Lagrange-Jacobi identity; Sundman's inequality; collision; regularization; Marchal-Chenciner's theorem; non-collision singularity; final motions; Chazy's classification; integrability; first integral; transverse heteroclinic intersection; monodromy group; differential Galois theory; Lindstedt series; von Zeipel series; small denominators; Birkhoff series; Lagrange and Laplace stability theorems; Arnold's theorem; quasi-periodic orbit; Nekhoroshev theorem; KAM theory; instability; symbolic dynamics

Domaines: Systèmes dynamiques [math.DS]

Jacques Fejoz. The three- and the N-body problem. Alessandra Celletti. Celestial Mechanics, Unesco-EOLSS, pp.1-30, 2013. ⟨hal-00915271⟩