The three- and the N-body problem
Résumé
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