The angular momentum of a relative equilibrium

Alain Chenciner

doi:10.3934/dcds.2013.33.1033

Article Contents

2013, Volume 33, Issue 3: 1033-1047. Doi: 10.3934/dcds.2013.33.1033

This issue Previous Article Variational approach to second species periodic solutions of Poincaré of the 3 body problem Next Article On the existence of bi--pyramidal central configurations of the $n+2$--body problem with an $n$--gon base

The angular momentum of a relative equilibrium

Alain Chenciner^1,

1.
ASD, IMCCE (UMR 8028), Observatoire de Paris, 77 avenue Denfert-Rochereau, 75014 Paris

Received: March 31, 2011

Revised: January 31, 2012

Published: February 2013

Abstract Related Papers Cited by

Abstract

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.

Keywords:

N-body problem,
Horn's problem.

Mathematics Subject Classification: Primary: 70F10, 15A18.

Citation:

Related Papers

Cited by

References

[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).