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March  2013, 33(3): 1033-1047. doi: 10.3934/dcds.2013.33.1033

The angular momentum of a relative equilibrium

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

Received  April 2011 Revised  February 2012 Published  October 2012

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.
Citation: Alain Chenciner. The angular momentum of a relative equilibrium. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1033-1047. doi: 10.3934/dcds.2013.33.1033
References:
 [1] A. Albouy, Integral manifolds of the $N$-body problem,, Inventiones Mathematicæ, 114 (1993), 463. doi: 10.1007/BF01232677. Google Scholar [2] A. Albouy, "Mutual Distances in Celestial Mechanics,", Lectures at Nankai Institute, (2004). Google Scholar [3] A. Albouy and A. Chenciner, Le problème des $n$ corps et les distances mutuelles,, Inventiones Mathematicæ, 131 (1998), 151. doi: 10.1007/s002220050200. Google Scholar [4] V. I. Arnold, "Mathematical methods of classical Mechanics,", Graduate Texts in Mathematics, (1989). Google Scholar [5] R. Bhatia, Linear algebra to quantum cohomology: The story of alfred Horn's inequalitites,, The American Mathematical Monthly, 108 (2001), 289. doi: 10.2307/2695237. Google Scholar [6] P. Birtea, I. Casu, T. Ratiu and M. Turhan, Stability of equilibria for the so$(4)$ free rigid body,, preprint, (). Google Scholar [7] A. Chenciner, The Lagrange reduction of the $N$-body problem: a survey,, preprint, (). Google Scholar [8] A. Chenciner, Symmetric 4-body balanced configurations and their relative equilibrium motions,, in preparation., (). Google Scholar [9] A. Chenciner and H. Jiménez-Pérez, Angular momentum and Horn's problem,, preprint, (). Google Scholar [10] W. Fulton, Eigenvalues of sums of hermitian matrices,, Séminaire Bourbaki, 1997/98 (1998). Google Scholar [11] W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus,, Bull. Amer. Math. Soc. (N. S.), 37 (2000), 209. Google Scholar [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. doi: 10.1353/ajm.2005.0005. Google Scholar [13] A. Knutson, The symplectic and algebraic geometry of Horn's problem,, Linear Algebra and its Applications, 319 (2000), 61. Google Scholar [14] A. Knutson and T. Tao, Honeycombs and sums of Hermitian matrices,, Notices of the AMS, 48 (2001). Google Scholar [15] H. B. Lawson Junior and M. L. Michelson, "Spin Geometry,", Princeton University Press (1989)., (1989). Google Scholar

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References:
 [1] A. Albouy, Integral manifolds of the $N$-body problem,, Inventiones Mathematicæ, 114 (1993), 463. doi: 10.1007/BF01232677. Google Scholar [2] A. Albouy, "Mutual Distances in Celestial Mechanics,", Lectures at Nankai Institute, (2004). Google Scholar [3] A. Albouy and A. Chenciner, Le problème des $n$ corps et les distances mutuelles,, Inventiones Mathematicæ, 131 (1998), 151. doi: 10.1007/s002220050200. Google Scholar [4] V. I. Arnold, "Mathematical methods of classical Mechanics,", Graduate Texts in Mathematics, (1989). Google Scholar [5] R. Bhatia, Linear algebra to quantum cohomology: The story of alfred Horn's inequalitites,, The American Mathematical Monthly, 108 (2001), 289. doi: 10.2307/2695237. Google Scholar [6] P. Birtea, I. Casu, T. Ratiu and M. Turhan, Stability of equilibria for the so$(4)$ free rigid body,, preprint, (). Google Scholar [7] A. Chenciner, The Lagrange reduction of the $N$-body problem: a survey,, preprint, (). Google Scholar [8] A. Chenciner, Symmetric 4-body balanced configurations and their relative equilibrium motions,, in preparation., (). Google Scholar [9] A. Chenciner and H. Jiménez-Pérez, Angular momentum and Horn's problem,, preprint, (). Google Scholar [10] W. Fulton, Eigenvalues of sums of hermitian matrices,, Séminaire Bourbaki, 1997/98 (1998). Google Scholar [11] W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus,, Bull. Amer. Math. Soc. (N. S.), 37 (2000), 209. Google Scholar [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. doi: 10.1353/ajm.2005.0005. Google Scholar [13] A. Knutson, The symplectic and algebraic geometry of Horn's problem,, Linear Algebra and its Applications, 319 (2000), 61. Google Scholar [14] A. Knutson and T. Tao, Honeycombs and sums of Hermitian matrices,, Notices of the AMS, 48 (2001). Google Scholar [15] H. B. Lawson Junior and M. L. Michelson, "Spin Geometry,", Princeton University Press (1989)., (1989). Google Scholar
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