Integrability of hamiltonian systems and differential Galois groups of higher variational equations
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 40 (2007) no. 6, pp. 845-884.
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     author = {Morales-Ruiz, Juan J. and Ramis, Jean-Pierre and Sim\'o, Carles},
     title = {Integrability of hamiltonian systems and differential {Galois} groups of higher variational equations},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {845--884},
     publisher = {Elsevier},
     volume = {Ser. 4, 40},
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     url = {http://www.numdam.org/articles/10.1016/j.ansens.2007.09.002/}
}
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Morales-Ruiz, Juan J.; Ramis, Jean-Pierre; Simó, Carles. Integrability of hamiltonian systems and differential Galois groups of higher variational equations. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 40 (2007) no. 6, pp. 845-884. doi : 10.1016/j.ansens.2007.09.002. http://www.numdam.org/articles/10.1016/j.ansens.2007.09.002/

[1] Abraham R., Marsden J.E., Foundations of Mechanics, second ed., Benjamin, 1978. | MR | Zbl

[2] Adler M., Van Moerbeke P., Vanhaecke P., Algebraic Integrability, Painlevé Geometry and Lie Algebras, Springer, Berlin, 2004. | MR | Zbl

[3] Almeida A., López-Castillo A., Stuchi T., Non-integrability proof of the frozen planetary atom configuration, J. Phys. A: Math. Gen. 36 (2003) 4805-4814. | MR | Zbl

[4] Almeida A., Stuchi T.J., The integrability of the anisotropic Stormer problem, Physica D 189 (2004) 219-233. | MR | Zbl

[5] Arribas M., Elipe A., Non Integrability of the motion of a particle around a massive straight segment, Phys. Lett. A 281 (2001) 142-148. | MR | Zbl

[6] Arribas M., Elipe A., Riaguas A., Non-integrability of anisotropic quasi-homogeneous Hamiltonian systems, Mech. Res. Commun. 30 (2003) 209-216. | MR | Zbl

[7] Artin M., On the solutions of analytic equations, Invent. Math. 5 (1968) 277-291. | MR | Zbl

[8] Atiyah M.F., Macdonald I.G., Introduction to Commutative Algebra, Addison-Wesley, London, 1969. | Zbl

[9] Audin M., Intégrabilité et non-intégrabilité de systèmes hamiltoniens (d'après S. Ziglin, J. Morales-Ruiz, J.-P. Ramis, …), Sém. Bourbaki (2000/01), Exp. no 884, Astérisque 282 (2002) 113-135. | Numdam | MR | Zbl

[10] Audin M., Les systèmes hamiltoniens et leur intégrabilité, Cours Spécialisés, Collection SMF, vol. 8, Société Mathématique de France, Marseille, 2001. | MR

[11] Audin M., Integrability of Hamiltonian systems, Bulletin E.M.S. (December 2003) 9-12.

[12] Audin M., Exemples de hamiltoniens non intégrables en mécanique analytique réelle, Ann. Fac. Sci. Toulouse Math. 12 (2003) 1-23. | Numdam | MR | Zbl

[13] Audin M., La réduction symplectique appliquée à la non-intégrabilité du problème du satellite, Ann. Fac. Sci. Toulouse Math. 12 (2003) 25-46. | Numdam | MR | Zbl

[14] Boucher D., Sur la non-intégrabilité du problème plan des trois corps de masses égales à un le long de la solution de Lagrange, C. R. Acad. Sci. Paris, Série I 331 (2000) 391-394. | MR | Zbl

[15] Boucher D., Sur les équations différentielles linéaires paramétrées, une application aux systèmes hamiltoniens, Thèse Univ. de Limoges, octobre 2000.

[16] Boucher D., Non complete integrability of a satellite in circular orbit, Portugaliae Mathematica (N.S.) 63 (2006) 69-89. | MR | Zbl

[17] Boucher D., Weil J.-A., Application of the J.-J. Morales and J.-P. Ramis theorem to test the non-complete integrability of the planar three-body problem, in: Fauvet F., Mitschi C. (Eds.), From Combinatorics to Dynamical Systems. Journées de calcul formel en l'honneur de Jean Thomann, March 22-23, 2002, IRMA Lectures in Mathematics and Theoretical Physics, vol. 3, IRMA, Strasbourg, 2003. | Zbl

[18] Bourbaki N., Algèbre commutative, chapitres 1 à 4, Springer-Verlag, 2006. | MR

[19] Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L., Griffiths P.A., Exterior Differential Systems, Springer-Verlag, Berlin, 1991. | MR | Zbl

[20] Casale G., Sur le groupoïde de Galois d'un feuilletage, Thèse, Toulouse 2004.

[21] Churchill R.C., Galoisian Obstructions to the Integrability of Hamiltonian Systems, The Kolchin Seminar in Differential Algebra, City College of New York, May, 1998.

[22] Churchill R.C., Differential algebraic techniques in Hamiltonian mechanics, in: Guo L., Keigher W.F., Cassidy P.J., Sit W.Y. (Eds.), Differential Algebra and Related Topics, World Scientific Publ., 2002, pp. 219-255. | MR

[23] Deligne P., Catégories tannakiennes, in: Grothendieck festschrift, vol. 2, Progress in Mathematics, vol. 87, Birkhäuser, 1990, pp. 111-196. | MR | Zbl

[24] Doubrovine B., Novikov S., Fomenko A., Géométrie Contemporaine, Méthodes et Applications, Deuxième partie, Géométrie et Topologie des Variétés, Éditions MIR, Moscou, 1979, (French translation 1982). | MR | Zbl

[25] Drach J., Essai sur une théorie générale de l'intégration et sur la classification des transcendantes, Ann. Sci. École Normale Sup., Sér. 3 15 (1898) 243-384. | JFM | Numdam | MR

[26] Drach J., Sur le problème logique de l'intégration des équations différentielles, Ann. Fac. Sci. Univ. Toulouse, Sér. 2 10 (1908) 393-472. | JFM | Numdam | MR

[27] Ferrer S., Mondéjar F., On the non-integrability of the Zeemann-Stark Hamiltonian system, Comm. Math. Phys. 208 (1999) 55-63. | Zbl

[28] Ferrer S., Mondéjar F., Non-integrability of the 3-D Hydrogen atom under motional Stark effect or circularly polarized microwave combined with magnetic fields, Phys. Lett. A 264 (1999) 74-83.

[29] Ferrer S., Mondéjar F., On the non-integrability of the generalized van der Waals Hamiltonian, J. Math. Phys. 41 (2000) 5445-5452. | MR | Zbl

[30] Fontich E., Simó C., Invariant manifolds for near identity differentiable maps and splitting of separatrices, Erg. Th. & Dyn. Systems 10 (1990) 319-346. | MR | Zbl

[31] Fontich E., Simó C., The splitting of separatrices for analytic diffeomorphisms, Erg. Th. & Dyn. Systems 10 (1990) 295-318. | MR | Zbl

[32] Grothendieck A., Techniques de construction en géométrie analytique VII, Séminaire Henri Cartan, 13-ème année 1960/61, vol. 14, W.A. Benjamin, 1967. | Numdam

[33] Hénon M., Heiles C., The applicability of the third integral of motion: Some numerical experiments, Astronom. J. 69 (1964) 73-79. | MR

[34] Irigoyen M., Simó C., Non-integrability of the J 2 problem, Celest. Mech. 55 (1993) 281-287. | Zbl

[35] Ito H., Non-integrability of the Hénon-Heiles system and a theorem of Ziglin, Koday Math. J. 8 (1985) 129-138. | Zbl

[36] Ito H., On the holonomy group associated with analytic continuations of solutions for integrable systems, Boletim da Soc. Brasil. de Matemática 21 (1990) 95-120. | MR | Zbl

[37] Juillard Tosel E., Non-integrabilité algébrique et méromorphe de problèmes de N corps, Thèse Univ. Paris VI, janvier 1999.

[38] Juillard Tosel E., Meromorphic parametric non-integrability; the inverse square potential, Arch. Rat. Mech. Anal. 152 (2000) 187-207. | MR | Zbl

[39] Kolar I., Michor P., Slovak J., Natural Operations in Differential Geometry, Springer-Verlag, 1991. | MR | Zbl

[40] Levine H.I., Singularities of differentiable mappings, in: Proceedings of Liverpool Singularities Symposium I, Lecture Notes in Math., vol. 192, 1971. | Zbl

[41] Llibre J., Simó C., Homoclinic phenomena in the three-body problem, J. Differential Equations 37 (1980) 444-465. | MR | Zbl

[42] Llibre J., Simó C., Oscillatory solutions in the planar restricted three-body problem, Math. Ann. 248 (1980) 153-184. | MR | Zbl

[43] Llibre J., Martínez R., Simó C., Transversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near L 2 in the Restricted Three Body Problem, J. Differential Equations 58 (1985) 104-156. | MR | Zbl

[44] Maciejewski A.J., Non-integrability of certain Hamiltonian systems, applications of the Morales-Ramis differential Galois extension of Ziglin theory, Banach Center Publications 58 (2002) 139-150. | Zbl

[45] Maciejewski A.J., Przybylska M., Non-integrability of restricted two-body problems in constant curvature spaces, Regul. Chaotic Dyn. 8 (4) (2003) 413-430. | MR | Zbl

[46] Maciejewski A.J., Przybylska M., Non-integrability of the problem of a rigid satellite in gravitational and magnetic fields, Celest. Mech. Dynam. Astron. 87 (2003) 317-351. | MR | Zbl

[47] Maciejewski A.J., Przybylska M., Non-integrability of the Suslov problem, J. Math. Phys. 45 (2004) 1065-1078. | Zbl

[48] Maciejewski A.J., Przybylska M., Non-integrability of the generalised two fixed centres problem, Celest. Mech. Dynam. Astron. 89 (2004) 145-164. | MR | Zbl

[49] Maciejewski A.J., Przybylska M., All meromorphically integrable 2D Hamiltonian systems with homogeneous potential of degree 3, Phys. Lett. A 327 (2004) 461-473. | MR

[50] Maciejewski A.J., Przybylska M., Differential Galois approach to the non-integrability of the heavy top problem, Ann. Fac. Sci. Toulouse Sér. 6 14 (2005) 123-160. | Numdam | MR | Zbl

[51] Maciejewski A.J., Przybylska M., Stachowiak T., Non integrability of Gross-Neveu systems, Physica D 201 (2005) 249-267. | Zbl

[52] Maciejewski A.J., Przybylska M., Weil J.-A., Non-integrability of the generalized spring-pendulum problem, J. Phys. A: Math. Gen. 37 (2004) 2579-2597. | MR | Zbl

[53] Maciejewski A.J., Strelcyn J.M., Szydlowski M., Non-integrability of Bianchi VIII Hamiltonian system, J. Math. Phys. 42 (2001) 1728-1743. | MR | Zbl

[54] Maciejewski A.J., Szydlowski M., Towards a description of complexity of the simplest cosmological systems, J. Phys. A: Math. Gen. 33 (2000) 9241-9254. | MR | Zbl

[55] Maciejewski A.J., Szydlowski M., Integrability and non-integrability of planar Hamiltonian systems of cosmological origin, J. Nonlinear Math. Phys. 8 (2001) 200-206. | MR | Zbl

[56] Malgrange B., Le groupoïde de Galois d'un feuilletage, L'enseignement mathématique 38 (2) (2001) 465-501. | MR | Zbl

[57] Malgrange B., On the non linear Galois theory, Chinese Ann. Math. Ser. B 23 (2) (2002) 219-226. | Zbl

[58] Martinet J., Singularités des fonctions et applications différentiables, Pontificia Universidade Católica do Rio de Janeiro (PUC), 1974. | MR | Zbl

[59] Morales-Ruiz J.J., Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Progress in Mathematics, vol. 179, Birkhäuser, 1999. | MR | Zbl

[60] Morales-Ruiz J.J., Kovalevskaya, Liapounov, Painlevé, Ziglin and the differential Galois theory, Regular Chaotic Dynam. 5 (2000) 251-272. | MR | Zbl

[61] Morales-Ruiz J.J., Peris J.M., On a Galoisian approach to the splitting of separatrices, Ann. Fac. Sci. Toulouse Math. 8 (1999) 125-141. | Numdam | MR | Zbl

[62] Morales-Ruiz J.J., Ramis J.P., Galoisian obstructions to integrability of Hamiltonian systems, Methods Appl. Anal. 8 (2001) 33-96. | MR

[63] Morales-Ruiz J.J., Ramis J.P., Galoisian obstructions to integrability of Hamiltonian systems II, Methods Appl. Anal. 8 (2001) 97-112.

[64] Morales-Ruiz J.J., Ramis J.P., A note on the non-integrability of some Hamiltonian systems with a homogeneous potential, Methods Appl. Anal. 8 (2001) 113-120. | MR

[65] Morales-Ruiz J.J., Simó C., Non-integrability criteria for Hamiltonians in the case of Lamé normal variational equations, J. Differential Equations 129 (1996) 111-135. | MR | Zbl

[66] Morales-Ruiz J.J., Simó C., Simon S., Algebraic proof of the non-integrability of Hill's problem, Ergod. Th. & Dynam. Sys. 25 (2005) 1237-1256. | MR | Zbl

[67] Moser J., Stable and Random Motions in Dynamical Systems, Annals of Mathematics Studies, Princeton Univ. Press, 1973. | MR | Zbl

[68] Nakagawa K., Direct construction of polynomial first integrals for Hamiltonian systems with a two-dimensional homogeneous polynomial potential, Dep. of Astronomical Science, The Graduate University for Advanced Study and the National Astronomical Observatory of Japan, Ph.D. Thesis, 2002.

[69] Nakagawa K., Yoshida H., A necessary condition for the integrability of homogeneous Hamiltonian systems with two degrees of freedom, J. Phys. A: Math. Gen. 34 (2001) 2137-2148. | MR | Zbl

[70] Neishtadt A.I., The separation of motions in systems with rapidly rotating phase, Prikladnaja Matematika i Mekhanika 48 (1984) 133-139. | MR | Zbl

[71] Poincaré H., Les Méthodes Nouvelles de la Mécanique Céleste, vol. I, Gauthiers-Villars, Paris, 1892. | Zbl

[72] Saenz A.W., Nonintegrability of the Dragt-Finn model of magnetic confinement: A Galoisian-group approach, Physica D 144 (2000) 37-43. | Zbl

[73] Serre J-P., Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier Grenoble (1956) 1-42. | Numdam | MR | Zbl

[74] Serre J-P., Espaces fibrés algébriques, in: Séminaire Claude Chavalley, 1958, tome 3, exp. no 1, pp. 1-37. | Numdam

[75] Simó C., Analytical and numerical computation of invariant manifolds, in: Benest D., Froeschlé C. (Eds.), Modern Methods in Celestial Mechanics, Editions Frontières, 1990, pp. 285-330, Also available at, http://www.maia.ub.es/dsg/.

[76] Simó C., Measuring the lack of integrability in the J 2 problem, in: Roy A.E. (Ed.), Predictability, Stability and Chaos in the N-Body Dynamical Systems, Plenum Press, 1991, pp. 305-309. | MR

[77] Terng C.L., Natural vector bundles and natural differential operators, Amer. J. Math. 100 (1978) 775-828. | MR | Zbl

[78] Tsygvintsev A., Sur l'absence d'une intégrale première méromorphe supplémentaire dans le problème plan des trois corps, C. R. Acad. Sci. Paris Série I 333 (2001) 241-244. | MR | Zbl

[79] Umemura H., Differential Galois theory of infinite dimension, Nagoya Math. J. 144 (1996) 59-135. | MR | Zbl

[80] Van Der Put M., Singer M., Galois Theory of Linear Differential Equations, Springer, Berlin, 2003. | MR | Zbl

[81] Vigo-Aguilar M.I., No integrabilidad del problema del satélite, Ph.D. Thesis, Departamento de Análisis Matemático y Matemática Aplicada, Universidad de Alicante, 1999.

[82] Yagasaki K., Galoisian obstructions to integrability and Melnikov criteria for chaos in two-degree-of-freedom Hamiltonian systems with saddle-centers, Nonlinearity 16 (2003) 2003-2013. | MR | Zbl

[83] Yagasaki K., Non-integrability of an infinity-degree-of-freedom model for unforced and undamped straight beams, J. Appl. Mech. 70 (2003) 732-738. | MR | Zbl

[84] Yoshida H., A new necessary condition for the integrability of Hamiltonian systems with a two dimensional homogeneous potential, Physica D 128 (1999) 53-69. | MR | Zbl

[85] Yoshida H., Justification of Painlevé Analysis for Hamiltonian systems by differential Galois theory, Physica A 228 (2000) 424-430. | MR

[86] Ziglin S.L., Branching of solutions and non-existence of first integrals in Hamiltonian mechanics I, Funct. Anal. Appl. 16 (1982) 181-189. | Zbl

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