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Moscow Mathematical Journal

Volume 13, Issue 4, October–December 2013  pp. 621–630.

Angular Momentum and Horn's Problem

Authors Alain Chenciner (1), Hugo Jiménez-Pérez (2)
Author institution: (1) Observatoire de Paris, IMCCE (UMR 8028), ASD 77, avenue Denfert-Rochereau, 75014 Paris, France and Department of Mathematics, University Paris 7
(2) Institut de Physique du Globe de Paris (UMR 7154), Department of Seismology 1, rue Jussieu, 75238 Paris Cedex 05, France


We prove a conjecture made by the first named author: Given an n-body central configuration X0 in the euclidean space E of dimension 2p, let Im ℱ be the set of decreasing real p-tuples (ν12,…,νp) such that {±iν1iν2,…,±iνp} is the spectrum of the angular momentum of some (periodic) relative equilibrium motion of X0 in E. Then Im ℱ is a convex polytope. The proof consists in showing that there exist two, generically (p−1)-dimensional, convex polytopes 𝒫1 and 𝒫2 in ℝp such that 𝒫1 ⊂ Im ℱ ⊂ 𝒫2 and that these two polytopes coincide.

𝒫1, introduced earlier in a paper by the first author, is the set of spectra corresponding to the hermitian structures J on E which are ``adapted'' to the symmetries of the inertia matrix S0; it is associated with Horn's problem for the sum of p×p real symmetric matrices with spectra σ and σ+ whose union is the spectrum of S0.

𝒫2 is the orthogonal projection onto the set of ``hermitian spectra'' of the polytope 𝒫 associated with Horn's problem for the sum of 2p×2p real symmetric matrices having each the same spectrum as S0$.

The equality 𝒫1=𝒫2 follows directly from a deep combinatorial lemma by S. Fomin, W. Fulton, C.K. Li, and Y.T. Poon, which characterizes those of the sums of two 2p×2p real symmetric matrices with the same spectrum which are hermitian for some hermitian structure.

2010 Mathematics Subject Classification. 70F10, 70E45, 15A18, 15B57

Keywords:  n-body problem, relative equilibrium, angular momentum, Horn's problem

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