2011/11/05 by Alain Chenciner, Chenciner, Alain · 1 citation
Engineering · Physics and Astronomy · #37J15 #70F10 #70H12 #70H33 #Astro and Planetary Science #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spacecraft Dynamics and Control #Stellar, planetary, and galactic studies
paper · doi:10.48550/arxiv.1111.1334
openalex publication_date 2011/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.