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On the existence of bi--pyramidal central configurations of the $n+2$--body problem with an $n$--gon base
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Variational approach to second species periodic solutions of Poincaré of the 3 body problem
The angular momentum of a relative equilibrium
| 1. | ASD, IMCCE (UMR 8028), Observatoire de Paris, 77 avenue Denfert-Rochereau, 75014 Paris |
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.
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. Google Scholar |
| [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, (). 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, 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). |
show all references
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. Google Scholar |
| [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, (). 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, 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). |
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