no. 21-22
Dynamical Systems/Mathematical Problems in Mechanics
Stability of relative equilibria and Morse index of central configurations
[Stabilité d'équilibre relatif et indice de Morse de configuration centrale]

Présenté par : Jean-Baptiste Leblond

Hu, Xijun1 ; Sun, Shanzhong2
1 Department of Mathematics, Shandong University, Jinan, Shandong, 250100, People's Republic of China
2 Department of Mathematics, Capital Normal University, Beijing, 100048, People's Republic of China
Comptes Rendus. Mathématique, Tome 347 (2009) no. 21-22, pp. 1309-1312.

Dans le problème plan des n corps, si l'indice de Morse ou la nullité d'une configuration centrale vue comme un point critique du potentiel newtonien restreint à la « sphère des formes » est impair, l'équilibre relatif correspondant est linéairement instable.

For the planar n-body problem, if the Morse index or the nullity of a central configuration as a critical point of Newton potential function restricted on the “shape sphere” is odd, then the relative equilibrium corresponding to the central configuration is linearly unstable.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.09.023
Hu, Xijun 1 ; Sun, Shanzhong 2

1 Department of Mathematics, Shandong University, Jinan, Shandong, 250100, People's Republic of China
2 Department of Mathematics, Capital Normal University, Beijing, 100048, People's Republic of China 
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Hu, Xijun; Sun, Shanzhong. Stability of relative equilibria and Morse index of central configurations. Comptes Rendus. Mathématique, Tome 347 (2009) no. 21-22, pp. 1309-1312. doi : 10.1016/j.crma.2009.09.023. http://www.numdam.org/articles/10.1016/j.crma.2009.09.023/

[1] Albouy, A.; Fu, Y.; Sun, S. Symmetry of planar four-body convex central configurations, Proc. R. Soc. A, Volume 464 (2008), pp. 1355-1365

[2] Long, Y. Index Theory for Symplectic Paths with Applications, Progress in Math., vol. 207, Birkhäuser, Basel, 2002

[3] McCord, C. Planar central configuration estimates in the N-body problem, Ergodic Theory Dyn. System., Volume 16 (1996), pp. 1059-1070

[4] Meyer, K.R.; Schmidt, D.S. Elliptic relative equilibria in the N-body problem, J. Differential Equations, Volume 214 (2005), pp. 256-298

[5] Moeckel, R. Celestial Mechanics, ICTP Lecture Notes, 1994

[6] Moeckel, R. Linear stability of relative equilibria with a dominant mass, J. Dynam. Differential Equations, Volume 6 (1994), pp. 37-51

[7] Palmore, J. Classifying relative equilibria, I, Bull. Amer. Math. Soc., Volume 79 (1973), pp. 904-908

[8] Roberts, G.E. Spectral instability of relative equilibria in the planar n-body problem, Nonlinearity, Volume 12 (1999), pp. 757-769

[9] Smale, S.; Smale, S. Topology and mechanics, II, Invent. Math., Volume 10 (1970), pp. 303-331

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