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.
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.
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@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://archive.numdam.org/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://archive.numdam.org/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://archive.numdam.org/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://archive.numdam.org/articles/10.1016/j.crma.2010.10.024/
[1] What is a completely integrable nonholonomic dynamical system?, Rep. Math. Phys., Volume 44 (1999) no. 1–2, pp. 29-35
[2] Extended integrability and bi-hamiltonian systems, Comm. Math. Phys., Volume 196 (1998) no. 1, pp. 19-51
[3] Differential Galois obstructions for non-commutative integrability, Phys. Lett. A, Volume 372 (2008) no. 33, pp. 5431-5435
[4] Necessary conditions for super-integrability of Hamiltonian systems, Phys. Lett. A, Volume 372 (2008) no. 34, pp. 5581-5587
[5] Galoisian obstructions to integrability of Hamiltonian systems, I and II, Methods Appl. Anal., Volume 8 (2001) no. 1, pp. 33-111
[6] 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] Singular complete integrability, Publ. Math. Inst. Hautes Etudes Sci., Volume 91 (2000), pp. 134-210
[8] Convergence versus integrability in Poincaré–Dulac normal forms, Math. Res. Lett. (2002)
[9] 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
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