The Lagrange reduction of the N-body problem, a survey

Abstract: In his fondamental "Essay on the 3-body problem", Lagrange, well before Jacobi's "reduction of the node", carries out the first complete reduction of symetries. Discovering the so-called homographic motions, he shows that they necessarily take place in a fixed plane. The true nature of this reduction is revealed if one considers the n-body problem in an euclidean space of arbitrary dimension. The actual dimension of the ambiant space then appears as a constraint, namely the angular momentum bivector's degeneracy. The main part of this survey is a detailed description of the results obtained in a joint paper with Alain Albouy published in french (Inventiones 1998): for a non homothetic homographic motion to exist, it is necessary that the space of motion be even dimensional. Two cases are possible: either the configuration is "central" (that is a critical point of the potential among configurations with a given moment of inertia) and the space where the motion takes place is endowed with an hermitian structure, or it is "balanced" (that is a critical point of the potential among configurations with a given inertia spectrum) and the motion is a new type, quasi-periodic, of relative equilibrium. Hip-Hops, which are substitutes to the non-existing homographic solutions with odd dimensional space of motion, are also discussed.