Article Contents
Article Contents

# The angular momentum of a relative equilibrium

• There are two main reasons why relative equilibria of $N$ point masses under the influence of Newton attraction are mathematically more interesting to study when space dimension is at least 4:
1) in a higher dimensional space, a relative equilibrium is determined not only by the initial configuration but also by the choice of a hermitian structure on the space where the motion takes place (see [3]); in particu\-lar, its angular momentum depends on this choice;
2) relative equilibria are not necessarily periodic: if the configuration is balanced but not central (see [3,2,7]), the motion is in general quasi-periodic.
In this exploratory paper we address the following question, which touches both aspects: what are the possible frequencies of the angular momentum of a given central (or balanced) configuration and at what values of these frequencies bifurcations from periodic to quasi-periodic relative equilibria do occur? We give a full answer for relative equilibrium motions in $R^4$ and conjecture that an analogous situation holds true for higher dimensions. A refinement of Horn's problem given in [12] plays an important role.
Mathematics Subject Classification: Primary: 70F10, 15A18.

 Citation:

•  [1] A. Albouy, Integral manifolds of the $N$-body problem, Inventiones Mathematicæ, 114 (1993), 463-488.doi: 10.1007/BF01232677. [2] A. Albouy, "Mutual Distances in Celestial Mechanics," Lectures at Nankai Institute, Tianjin, China, 2004. [3] A. Albouy and A. Chenciner, Le problème des $n$ corps et les distances mutuelles, Inventiones Mathematicæ, 131 (1998), 151-184.doi: 10.1007/s002220050200. [4] V. I. Arnold, "Mathematical methods of classical Mechanics," Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. [5] R. Bhatia, Linear algebra to quantum cohomology: The story of alfred Horn's inequalitites, The American Mathematical Monthly, 108 (2001), 289-318.doi: 10.2307/2695237. [6] P. Birtea, I. Casu, T. Ratiu and M. Turhan, Stability of equilibria for the so$(4)$ free rigid body, preprint, arXiv:0812.3415. [7] A. Chenciner, The Lagrange reduction of the $N$-body problem: a survey, preprint, arXiv:1111.1334, submitted to Acta Mathematica Vietnamica. [8] A. Chenciner, Symmetric 4-body balanced configurations and their relative equilibrium motions, in preparation. [9] A. Chenciner and H. Jiménez-Pérez, Angular momentum and Horn's problem, preprint, arXiv:1110.5030, submitted to the Moscow Mathematical Journal. [10] W. Fulton, Eigenvalues of sums of hermitian matrices, Séminaire Bourbaki, exposé, 1997/98 (1998), 255–-269. [11] W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. (N. S.), 37 (2000), 209-249. [12] S. Fomin, W. Fulton, C. K. Li and Y. T. Poon, Eigenvalues, singular values, and Little wood-Richardson coefficients, Amer. J. Math., 127 (2005), 101-127.doi: 10.1353/ajm.2005.0005. [13] A. Knutson, The symplectic and algebraic geometry of Horn's problem, Linear Algebra and its Applications, 319 (2000), 61-81. [14] A. Knutson and T. Tao, Honeycombs and sums of Hermitian matrices, Notices of the AMS, Soc., 48 (2001), 175-–186. [15] H. B. Lawson Junior and M. L. Michelson, "Spin Geometry," Princeton University Press (1989).