# 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

**Summary:**

We prove a conjecture made by the first named
author: Given an *n*-body central configuration *X*_{0} in the
euclidean space *E* of dimension 2*p*, let Im ℱ be the set of
decreasing real *p*-tuples
(ν_{1},ν_{2},…,ν_{p}) such that
{±iν_{1},±*i*ν_{2},…,±*i*ν_{p}} is the spectrum of the
angular momentum of some (periodic) relative equilibrium motion of
*X*_{0} in *E*. Then Im ℱ is a convex polytope. The proof
consists in showing that there exist two, generically
(*p*−1)-dimensional, convex polytopes 𝒫_{1} and 𝒫_{2} in
ℝ^{p} such that 𝒫_{1} ⊂ Im ℱ ⊂ 𝒫_{2} and that these two polytopes coincide.

𝒫_{1}, 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 *S*_{0}; 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 *S*_{0}.

𝒫_{2} is the orthogonal projection onto the set of ``hermitian spectra'' of the polytope
𝒫 associated with Horn's problem for the sum of 2*p*×2*p* real symmetric matrices having each the same spectrum as *S*_{0}$.

The equality 𝒫_{1}=𝒫_{2} 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 2*p*×2*p* 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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