In this article we give an obstruction to integrability by quadratures of an ordinary differential equation on the differential Galois group of variational equations of any order along a particular solution. In Hamiltonian situation the condition on the Galois group gives Morales-Ramis-Simó theorem. The main tools used are Malgrange pseudogroup of a vector field and Artin approximation theorem.
Dans cet article, nous montrons que les équations variationnelles le long d’une solution d’une équation différentielle intégrable par quadratures ont un groupe de Galois différentielle virtuellement résoluble. Dans le cas particulier des systèmes hamiltoniens intégrables au sens de Liouville la preuve redonne le théorème de Morales-Ramis-Simó. La preuve consiste à montrer que le groupe de Galois de l’équation variationnelle est un quotient d’un sous groupe d’un groupe d’isotropie du pseudogroupe de Malgrange de l’équation non linéaire. On relie ensuite les propriétés de ce groupe d’isotropie en un point spécial à celles du groupe d’isotropie au point générique en utilisant le théorème d’approximation d’Artin.
Keywords: Differential Galois theory, variational equation, integrability
Mot clés : Théorie de Galois différentielle, équations variationnelles, intégrabilité
@article{AIF_2009__59_7_2593_0, author = {Casale, Guy}, title = {Morales-Ramis {Theorems} via {Malgrange} pseudogroup}, journal = {Annales de l'Institut Fourier}, pages = {2593--2610}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {59}, number = {7}, year = {2009}, doi = {10.5802/aif.2501}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2501/} }
TY - JOUR AU - Casale, Guy TI - Morales-Ramis Theorems via Malgrange pseudogroup JO - Annales de l'Institut Fourier PY - 2009 SP - 2593 EP - 2610 VL - 59 IS - 7 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2501/ DO - 10.5802/aif.2501 LA - en ID - AIF_2009__59_7_2593_0 ER -
%0 Journal Article %A Casale, Guy %T Morales-Ramis Theorems via Malgrange pseudogroup %J Annales de l'Institut Fourier %D 2009 %P 2593-2610 %V 59 %N 7 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2501/ %R 10.5802/aif.2501 %G en %F AIF_2009__59_7_2593_0
Casale, Guy. Morales-Ramis Theorems via Malgrange pseudogroup. Annales de l'Institut Fourier, Volume 59 (2009) no. 7, pp. 2593-2610. doi : 10.5802/aif.2501. http://www.numdam.org/articles/10.5802/aif.2501/
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