Morales-Ramis Theorems via Malgrange pseudogroup
[Les théorèmes de Morales-Ramis via le pseudo-groupe de Malgrange]
Casale, Guy
Annales de l'Institut Fourier, Tome 59 (2009), p. 2593-2610 / Harvested from Numdam
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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.

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.

Publié le : 2009-01-01
DOI : https://doi.org/10.5802/aif.2501
Classification:  53A55,  34A34
Mots clés: Théorie de Galois différentielle, équations variationnelles, intégrabilité 
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     author = {Casale, Guy},
     title = {Morales-Ramis Theorems  via Malgrange pseudogroup},
     journal = {Annales de l'Institut Fourier},
     volume = {59},
     year = {2009},
     pages = {2593-2610},
     doi = {10.5802/aif.2501},
     zbl = {pre05689400},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2009__59_7_2593_0}
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Casale, Guy. Morales-Ramis Theorems  via Malgrange pseudogroup. Annales de l'Institut Fourier, Tome 59 (2009) pp. 2593-2610. doi : 10.5802/aif.2501. http://gdmltest.u-ga.fr/item/AIF_2009__59_7_2593_0/

[1] Adler, Mark; Van Moerbeke, Pierre; Vanhaecke, Pol Algebraic integrability, Painlevé geometry and Lie algebras, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Tome 47 (2004) | MR 2095251

[2] Artin, M. On the solutions of analytic equations, Invent. Math., Tome 5 (1968), pp. 277-291 | Article | MR 232018 | Zbl 0172.05301

[3] Audin, Michèle Les systèmes hamiltoniens et leur intégrabilité, Société Mathématique de France, Paris, Cours Spécialisés [Specialized Courses], Tome 8 (2001) | MR 1972063 | Zbl 1144.37001

[4] Ayoul, M.; Zung, N. T. Galoisian obstruction to non-Hamiltonian integrability (2009) (arXiv:0901.4586)

[5] Bronstein, Manuel; Lafaille, Sébastien Solutions of linear ordinary differential equations in terms of special functions, Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, ACM, New York (2002), p. 23-28 (electronic) | MR 2035229 | Zbl 1072.68652

[6] Canalis, M.; Ramis, J.-P.; Rouchon, P.; Weil, J.-A. Calculations on the Lorenz system: Variational equation, Bessel dynamics (2001) (MAPLE worksheet available on http://perso.univ-rennes1.fr/guy.casale/ANR/ANR_html/publications.html)

[7] Casale, G. Une preuve galoisienne de l’irréductibilité au sens de Nishioka-Umemura de la 1ère équation de Painlevé, Astérisque, Soc. Math. de France, Tome 324 (2009), pp. 83-100 (Differential Equation and Singularities, 60th years of J.-M. Aroca)

[8] Casale, Guy; Roques, Julien Dynamics of rational symplectic mappings and difference Galois theory, Int. Math. Res. Not. IMRN (2008), pp. Art. ID rnn 103, 23 | MR 2439539 | Zbl 1172.37022

[9] Churchill, Richard C.; Rod, David L. On the determination of Ziglin monodromy groups, SIAM J. Math. Anal., Tome 22 (1991) no. 6, pp. 1790-1802 | Article | MR 1129412 | Zbl 0739.58018

[10] Churchill, Richard C.; Rod, David L.; Singer, M. F. Group-theoretic obstructions to integrability, Ergodic Theory Dynam. Systems, Tome 15 (1995) no. 1, pp. 15-48 | Article | Zbl 0824.58021

[11] Gabriel, Pierre Construction de préschémas quotient, Schémas en Groupes (Sém. Géométrie Algébrique, Inst. Hautes Études Sci., 1963/64), Fasc. 2a, Exposé 5, Inst. Hautes Études Sci., Paris (1963), pp. 37 | MR 257095

[12] Guillemin, Victor W.; Sternberg, Shlomo An algebraic model of transitive differential geometry, Bull. Amer. Math. Soc., Tome 70 (1964), pp. 16-47 | Article | MR 170295 | Zbl 0121.38801

[13] Ito, Hidekazu On the holonomy group associated with analytic continuations of solutions for integrable systems, Bol. Soc. Brasil. Mat. (N.S.), Tome 21 (1990) no. 1, pp. 95-120 | Article | MR 1139560 | Zbl 0762.58015

[14] Maciejewski, Andrzej J.; Przybylska, Maria Differential Galois obstructions for non-commutative integrability, Phys. Lett. A, Tome 372 (2008) no. 33, pp. 5431-5435 | Article | MR 2439693

[15] Mackenzie, K. Lie groupoids and Lie algebroids in differential geometry, Cambridge University Press, Cambridge, London Mathematical Society Lecture Note Series, Tome 124 (1987) | MR 896907 | Zbl 0683.53029

[16] Malgrange, Bernard Le groupoïde de Galois d’un feuilletage, Essays on geometry and related topics, Vol. 1, 2, Enseignement Math., Geneva (Monogr. Enseign. Math.) Tome 38 (2001), pp. 465-501 | MR 1929336 | Zbl 1033.32020

[17] Malgrange, Bernard On nonlinear differential Galois theory, Chinese Ann. Math. Ser. B, Tome 23 (2002) no. 2, pp. 219-226 (Dedicated to the memory of Jacques-Louis Lions) | Article | MR 1924138 | Zbl 1009.12005

[18] Malgrange, Bernard Personal discutions (2007)

[19] Morales, J. J.; Simó, C. Picard-Vessiot theory and Ziglin’s theorem, J. Differential Equations, Tome 107 (1994) no. 1, pp. 140-162 | Article | MR 1260852 | Zbl 0799.58035

[20] Morales-Ruiz, Juan J. A remark about the Painlevé transcendents, Théories asymptotiques et équations de Painlevé, Soc. Math. France, Paris (Sémin. Congr.) Tome 14 (2006), pp. 229-235 | MR 2353467 | Zbl 1140.37016

[21] Morales-Ruiz, Juan J.; Ramis, Jean Pierre Galoisian obstructions to integrability of Hamiltonian systems. I, II, Methods Appl. Anal., Tome 8 (2001) no. 1, p. 33-95, 97–111 | MR 1867495 | Zbl 1140.37354

[22] Morales-Ruiz, Juan J.; Ramis, Jean-Pierre; Simó, Carles Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Ann. Sci. École Norm. Sup. (4), Tome 40 (2007) no. 6, pp. 845-884 | Article | Numdam | MR 2419851 | Zbl 1144.37023

[23] Noumi, Masatoshi; Okamoto, Kazuo Irreducibility of the second and the fourth Painlevé equations, Funkcial. Ekvac., Tome 40 (1997) no. 1, pp. 139-163 http://www.math.kobe-u.ac.jp/~fe/xml/mr1454468.xml | MR 1454468 | Zbl 0881.34052

[24] Pommaret, J.-F. Differential Galois theory, Gordon & Breach Science Publishers, New York, Mathematics and its Applications, Tome 15 (1983) | MR 720863 | Zbl 0539.12013

[25] Przybylska, Maria Differential Galois obstructions for integrability of homogeneous Newton equations, J. Math. Phys., Tome 49 (2008) no. 2, pp. 022701, 40 | Article | MR 2392854 | Zbl 1153.81420

[26] Ritt, Joseph Fels Differential algebra, Dover Publications Inc., New York (1966) | MR 201431 | Zbl 0141.03801

[27] Umemura, Hiroshi; Watanabe, Humihiko Solutions of the second and fourth Painlevé equations. I, Nagoya Math. J., Tome 148 (1997), pp. 151-198 | MR 1492945 | Zbl 0934.33029

[28] Ziglin, S. L. Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics. I, Funct. Anal. Appl., Tome 16 (1983), pp. 181-189 (Translation from Funkts. Anal. Prilozh. 16, No.3, 30–41 (Russian) (1982)) | Article | Zbl 0524.58015

[29] Zung, Nguyen Tien Convergence versus integrability in Poincaré-Dulac normal form., Math. Res. Lett., Tome 9 (2002) no. 2-3, pp. 217-228 | MR 1909639 | Zbl 1019.34084