Geodesic
Dynamical Systems/Mathematical Problems in Mechanics
Galoisian obstructions to non-Hamiltonian integrability
[Obstructions galoisiennes à l'intégrabilité non-hamiltonien]

Présenté par : Jean-Pierre Ramis

Ayoul, Michaël1 ; Zung, Nguyen Tien1
1 Institut de mathématiques de Toulouse, UMR 5219 CNRS, université Toulouse III, 118 route de Narbonne, 31000 Toulouse, France
Comptes Rendus. Mathématique, Tome 348 (2010) no. 23-24, pp. 1323-1326.

Voir la notice de l'article provenant de la source Numdam

We show that the main theorem of Morales, Ramis and Simo (2007) [6] about Galoisian obstructions to meromorphic integrability of Hamiltonian systems can be naturally extended to the non-Hamiltonian case. Namely, if a dynamical system is meromorphically integrable in the non-Hamiltonian sense, then the differential Galois groups of the variational equations (of any order) along its solutions must be virtually Abelian.

Nous montrons la version non-hamiltonienne du théorème de Morales, Ramis et Simo (2007) [6]. Plus précisément, si un système dynamique est méromorphiquement intégrable au sens non-hamiltonien, alors tous les groupes de Galois différentiels des équations variationelles d'ordre arbitraire le long de ses solutions doivent être virtuellement abéliens.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.10.024

Ayoul, Michaël 1 ; Zung, Nguyen Tien 1

1 Institut de mathématiques de Toulouse, UMR 5219 CNRS, université Toulouse III, 118 route de Narbonne, 31000 Toulouse, France 
@article{CRMATH_2010__348_23-24_1323_0,
     author = {Ayoul, Micha\"el and Zung, Nguyen Tien},
     title = {Galoisian obstructions to {non-Hamiltonian} integrability},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1323--1326},
     publisher = {Elsevier},
     volume = {348},
     number = {23-24},
     year = {2010},
     doi = {10.1016/j.crma.2010.10.024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2010.10.024/}
}
TY  - JOUR
AU  - Ayoul, Michaël
AU  - Zung, Nguyen Tien
TI  - Galoisian obstructions to non-Hamiltonian integrability
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 1323
EP  - 1326
VL  - 348
IS  - 23-24
PB  - Elsevier
UR  - http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2010.10.024/
DO  - 10.1016/j.crma.2010.10.024
LA  - en
ID  - CRMATH_2010__348_23-24_1323_0
ER  - 
%0 Journal Article
%A Ayoul, Michaël
%A Zung, Nguyen Tien
%T Galoisian obstructions to non-Hamiltonian integrability
%J Comptes Rendus. Mathématique
%D 2010
%P 1323-1326
%V 348
%N 23-24
%I Elsevier
%U http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2010.10.024/
%R 10.1016/j.crma.2010.10.024
%G en
%F CRMATH_2010__348_23-24_1323_0
Ayoul, Michaël; Zung, Nguyen Tien. Galoisian obstructions to non-Hamiltonian integrability. Comptes Rendus. Mathématique, Tome 348 (2010) no. 23-24, pp. 1323-1326. doi : 10.1016/j.crma.2010.10.024. http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2010.10.024/

[1] Bates, L.; Cushman, R. What is a completely integrable nonholonomic dynamical system?, Rep. Math. Phys., Volume 44 (1999) no. 1–2, pp. 29-35

[2] Bogoyavlenskij, O.I. Extended integrability and bi-hamiltonian systems, Comm. Math. Phys., Volume 196 (1998) no. 1, pp. 19-51

[3] Maciejewski, A.; Przybylska, M. Differential Galois obstructions for non-commutative integrability, Phys. Lett. A, Volume 372 (2008) no. 33, pp. 5431-5435

[4] Maciejewski, A.; Przybylska, M.; Yoshida, H. Necessary conditions for super-integrability of Hamiltonian systems, Phys. Lett. A, Volume 372 (2008) no. 34, pp. 5581-5587

[5] Morales-Ruiz, J.; Ramis, J.-P. Galoisian obstructions to integrability of Hamiltonian systems, I and II, Methods Appl. Anal., Volume 8 (2001) no. 1, pp. 33-111

[6] Morales-Ruiz, J.; Ramis, J.-P.; Simo, C. Integrability of Hamiltonian systems and differential Galois groups of higher order variational equations, Ann. Sci. Ec. Norm. Super., Volume 40 (2007) no. 6, pp. 845-884

[7] Stolovitch, L. Singular complete integrability, Publ. Math. Inst. Hautes Etudes Sci., Volume 91 (2000), pp. 134-210

[8] Zung, N.T. Convergence versus integrability in Poincaré–Dulac normal forms, Math. Res. Lett. (2002)

[9] Zung, N.T. Torus actions and integrable systems (Bolsinov, A.V.; Fomenko, A.T.; Oshemkov, A.A., eds.), Topological Methods in the Theory of Integrable Systems, Cambridge Sci. Publ., 2006, pp. 289-328 | arXiv

Cité par Sources :