@article{ASENS_2007_4_40_6_845_0, 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}, volume = {40}, year = {2007}, pages = {845-884}, doi = {10.1016/j.ansens.2007.09.002}, mrnumber = {2419851}, zbl = {1144.37023}, language = {en}, url = {http://dml.mathdoc.fr/item/ASENS_2007_4_40_6_845_0} }
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, Tome 40 (2007) pp. 845-884. doi : 10.1016/j.ansens.2007.09.002. http://gdmltest.u-ga.fr/item/ASENS_2007_4_40_6_845_0/
[1] Foundations of Mechanics, second ed., Benjamin, 1978. | MR 515141 | Zbl 0393.70001
, ,[2] Algebraic Integrability, Painlevé Geometry and Lie Algebras, Springer, Berlin, 2004. | MR 2095251 | Zbl 1083.37001
, , ,[3] Non-integrability proof of the frozen planetary atom configuration, J. Phys. A: Math. Gen. 36 (2003) 4805-4814. | MR 1984009 | Zbl 1056.70006
, , ,[4] The integrability of the anisotropic Stormer problem, Physica D 189 (2004) 219-233. | MR 2065518 | Zbl 1098.70513
, ,[5] Non Integrability of the motion of a particle around a massive straight segment, Phys. Lett. A 281 (2001) 142-148. | MR 1822631 | Zbl 1009.70017
, ,[6] Non-integrability of anisotropic quasi-homogeneous Hamiltonian systems, Mech. Res. Commun. 30 (2003) 209-216. | MR 1965203 | Zbl 1038.70012
, , ,[7] On the solutions of analytic equations, Invent. Math. 5 (1968) 277-291. | MR 232018 | Zbl 0172.05301
,[8] Introduction to Commutative Algebra, Addison-Wesley, London, 1969. | Zbl 0175.03601
, ,[9] 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 1975177 | Zbl 1050.37028
,[10] Les systèmes hamiltoniens et leur intégrabilité, Cours Spécialisés, Collection SMF, vol. 8, Société Mathématique de France, Marseille, 2001. | MR 1972063 | Zbl pre01682500
,[11] Integrability of Hamiltonian systems, Bulletin E.M.S. (December 2003) 9-12.
,[12] Exemples de hamiltoniens non intégrables en mécanique analytique réelle, Ann. Fac. Sci. Toulouse Math. 12 (2003) 1-23. | Numdam | MR 2124073 | Zbl 1042.37039
,[13] La réduction symplectique appliquée à la non-intégrabilité du problème du satellite, Ann. Fac. Sci. Toulouse Math. 12 (2003) 25-46. | Numdam | MR 2124074 | Zbl 1042.37040
,[14] 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 1784920 | Zbl 0980.70008
,[15] Boucher D., Sur les équations différentielles linéaires paramétrées, une application aux systèmes hamiltoniens, Thèse Univ. de Limoges, octobre 2000.
[16] Non complete integrability of a satellite in circular orbit, Portugaliae Mathematica (N.S.) 63 (2006) 69-89. | MR 2211962 | Zbl 1089.70012
,[17] Application of the J.-J. Morales and J.-P. Ramis theorem to test the non-complete integrability of the planar three-body problem, in: , (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 1039.37034
, ,[18] Algèbre commutative, chapitres 1 à 4, Springer-Verlag, 2006. | MR 2284892
,[19] Exterior Differential Systems, Springer-Verlag, Berlin, 1991. | MR 1083148 | Zbl 0726.58002
, , , , ,[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] Differential algebraic techniques in Hamiltonian mechanics, in: , , , (Eds.), Differential Algebra and Related Topics, World Scientific Publ., 2002, pp. 219-255. | MR 1921701 | Zbl pre01824075
,[23] Catégories tannakiennes, in: Grothendieck festschrift, vol. 2, Progress in Mathematics, vol. 87, Birkhäuser, 1990, pp. 111-196. | MR 1106898 | Zbl 0727.14010
,[24] Géométrie Contemporaine, Méthodes et Applications, Deuxième partie, Géométrie et Topologie des Variétés, Éditions MIR, Moscou, 1979, (French translation 1982). | MR 728386 | Zbl 0502.53003
, , ,[25] 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 29.0349.06 | Numdam | MR 1508959
,[26] Sur le problème logique de l'intégration des équations différentielles, Ann. Fac. Sci. Univ. Toulouse, Sér. 2 10 (1908) 393-472. | JFM 41.0416.03 | Numdam | MR 1508305
,[27] On the non-integrability of the Zeemann-Stark Hamiltonian system, Comm. Math. Phys. 208 (1999) 55-63. | Zbl 0988.37072
, ,[28] 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] On the non-integrability of the generalized van der Waals Hamiltonian, J. Math. Phys. 41 (2000) 5445-5452. | MR 1770965 | Zbl 0997.37031
, ,[30] Invariant manifolds for near identity differentiable maps and splitting of separatrices, Erg. Th. & Dyn. Systems 10 (1990) 319-346. | MR 1062761 | Zbl 0706.58060
, ,[31] The splitting of separatrices for analytic diffeomorphisms, Erg. Th. & Dyn. Systems 10 (1990) 295-318. | MR 1062760 | Zbl 0706.58061
, ,[32] Techniques de construction en géométrie analytique VII, Séminaire Henri Cartan, 13-ème année 1960/61, vol. 14, W.A. Benjamin, 1967. | Numdam
,[33] The applicability of the third integral of motion: Some numerical experiments, Astronom. J. 69 (1964) 73-79. | MR 158746
, ,[34] Non-integrability of the problem, Celest. Mech. 55 (1993) 281-287. | Zbl 0767.70015
, ,[35] Non-integrability of the Hénon-Heiles system and a theorem of Ziglin, Koday Math. J. 8 (1985) 129-138. | Zbl 0577.58016
,[36] On the holonomy group associated with analytic continuations of solutions for integrable systems, Boletim da Soc. Brasil. de Matemática 21 (1990) 95-120. | MR 1139560 | Zbl 0762.58015
,[37] Juillard Tosel E., Non-integrabilité algébrique et méromorphe de problèmes de N corps, Thèse Univ. Paris VI, janvier 1999.
[38] Meromorphic parametric non-integrability; the inverse square potential, Arch. Rat. Mech. Anal. 152 (2000) 187-207. | MR 1764944 | Zbl 0963.70010
,[39] Natural Operations in Differential Geometry, Springer-Verlag, 1991. | MR 1202431 | Zbl 0782.53013
, , ,[40] Singularities of differentiable mappings, in: Proceedings of Liverpool Singularities Symposium I, Lecture Notes in Math., vol. 192, 1971. | Zbl 0216.45803
,[41] Homoclinic phenomena in the three-body problem, J. Differential Equations 37 (1980) 444-465. | MR 590002 | Zbl 0445.70005
, ,[42] Oscillatory solutions in the planar restricted three-body problem, Math. Ann. 248 (1980) 153-184. | MR 573346 | Zbl 0505.70010
, ,[43] Transversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near in the Restricted Three Body Problem, J. Differential Equations 58 (1985) 104-156. | MR 791842 | Zbl 0594.70013
, , ,[44] 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 1033.37028
,[45] Non-integrability of restricted two-body problems in constant curvature spaces, Regul. Chaotic Dyn. 8 (4) (2003) 413-430. | MR 2023045 | Zbl 1048.37052
, ,[46] Non-integrability of the problem of a rigid satellite in gravitational and magnetic fields, Celest. Mech. Dynam. Astron. 87 (2003) 317-351. | MR 2027746 | Zbl 1106.70330
, ,[47] Non-integrability of the Suslov problem, J. Math. Phys. 45 (2004) 1065-1078. | Zbl 1070.70003
, ,[48] Non-integrability of the generalised two fixed centres problem, Celest. Mech. Dynam. Astron. 89 (2004) 145-164. | MR 2086188 | Zbl 1132.70006
, ,[49] All meromorphically integrable 2D Hamiltonian systems with homogeneous potential of degree 3, Phys. Lett. A 327 (2004) 461-473. | MR 2086138 | Zbl pre05250970
, ,[50] Differential Galois approach to the non-integrability of the heavy top problem, Ann. Fac. Sci. Toulouse Sér. 6 14 (2005) 123-160. | Numdam | MR 2118036 | Zbl 1089.70002
, ,[51] Non integrability of Gross-Neveu systems, Physica D 201 (2005) 249-267. | Zbl 1088.37028
, , ,[52] Non-integrability of the generalized spring-pendulum problem, J. Phys. A: Math. Gen. 37 (2004) 2579-2597. | MR 2047228 | Zbl 1076.70010
, , ,[53] Non-integrability of Bianchi VIII Hamiltonian system, J. Math. Phys. 42 (2001) 1728-1743. | MR 1820428 | Zbl 1005.37057
, , ,[54] Towards a description of complexity of the simplest cosmological systems, J. Phys. A: Math. Gen. 33 (2000) 9241-9254. | MR 1804891 | Zbl 0970.83011
, ,[55] Integrability and non-integrability of planar Hamiltonian systems of cosmological origin, J. Nonlinear Math. Phys. 8 (2001) 200-206. | MR 1821531 | Zbl 0979.83044
, ,[56] Le groupoïde de Galois d'un feuilletage, L'enseignement mathématique 38 (2) (2001) 465-501. | MR 1929336 | Zbl 1033.32020
,[57] On the non linear Galois theory, Chinese Ann. Math. Ser. B 23 (2) (2002) 219-226. | Zbl 1009.12005
,[58] Singularités des fonctions et applications différentiables, Pontificia Universidade Católica do Rio de Janeiro (PUC), 1974. | MR 464292 | Zbl 0389.58005
,[59] Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Progress in Mathematics, vol. 179, Birkhäuser, 1999. | MR 1713573 | Zbl 0934.12003
,[60] Kovalevskaya, Liapounov, Painlevé, Ziglin and the differential Galois theory, Regular Chaotic Dynam. 5 (2000) 251-272. | MR 1789476 | Zbl 0984.34075
,[61] On a Galoisian approach to the splitting of separatrices, Ann. Fac. Sci. Toulouse Math. 8 (1999) 125-141. | Numdam | MR 1721562 | Zbl 0971.34076
, ,[62] Galoisian obstructions to integrability of Hamiltonian systems, Methods Appl. Anal. 8 (2001) 33-96. | MR 1867495 | Zbl pre01757335
, ,[63] Galoisian obstructions to integrability of Hamiltonian systems II, Methods Appl. Anal. 8 (2001) 97-112. | Zbl pre01823952
, ,[64] A note on the non-integrability of some Hamiltonian systems with a homogeneous potential, Methods Appl. Anal. 8 (2001) 113-120. | MR 1867496 | Zbl pre01757336
, ,[65] Non-integrability criteria for Hamiltonians in the case of Lamé normal variational equations, J. Differential Equations 129 (1996) 111-135. | MR 1400798 | Zbl 0866.58034
, ,[66] Algebraic proof of the non-integrability of Hill's problem, Ergod. Th. & Dynam. Sys. 25 (2005) 1237-1256. | MR 2158404 | Zbl 1078.70014
, , ,[67] Stable and Random Motions in Dynamical Systems, Annals of Mathematics Studies, Princeton Univ. Press, 1973. | MR 442980 | Zbl 0271.70009
,[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] 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 1831282 | Zbl 1054.37037
, ,[70] The separation of motions in systems with rapidly rotating phase, Prikladnaja Matematika i Mekhanika 48 (1984) 133-139. | MR 802878 | Zbl 0571.70022
,[71] Les Méthodes Nouvelles de la Mécanique Céleste, vol. I, Gauthiers-Villars, Paris, 1892. | Zbl 0651.70002
,[72] Nonintegrability of the Dragt-Finn model of magnetic confinement: A Galoisian-group approach, Physica D 144 (2000) 37-43. | Zbl 0958.37029
,[73] Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier Grenoble (1956) 1-42. | Numdam | MR 82175 | Zbl 0075.30401
,[74] Serre J-P., Espaces fibrés algébriques, in: Séminaire Claude Chavalley, 1958, tome 3, exp. no 1, pp. 1-37. | Numdam
[75] Analytical and numerical computation of invariant manifolds, in: , (Eds.), Modern Methods in Celestial Mechanics, Editions Frontières, 1990, pp. 285-330, Also available at, http://www.maia.ub.es/dsg/.
,[76] Measuring the lack of integrability in the problem, in: (Ed.), Predictability, Stability and Chaos in the N-Body Dynamical Systems, Plenum Press, 1991, pp. 305-309. | MR 1210981
,[77] Natural vector bundles and natural differential operators, Amer. J. Math. 100 (1978) 775-828. | MR 509074 | Zbl 0422.58001
,[78] 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 1847358 | Zbl 0964.70011
,[79] Differential Galois theory of infinite dimension, Nagoya Math. J. 144 (1996) 59-135. | MR 1425592 | Zbl 0878.12002
,[80] Galois Theory of Linear Differential Equations, Springer, Berlin, 2003. | MR 1960772 | Zbl 1036.12008
, ,[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] 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 2012852 | Zbl 1070.37038
,[83] Non-integrability of an infinity-degree-of-freedom model for unforced and undamped straight beams, J. Appl. Mech. 70 (2003) 732-738. | MR 2011650 | Zbl 1110.74769
,[84] A new necessary condition for the integrability of Hamiltonian systems with a two dimensional homogeneous potential, Physica D 128 (1999) 53-69. | MR 1685248 | Zbl 0945.37015
,[85] Justification of Painlevé Analysis for Hamiltonian systems by differential Galois theory, Physica A 228 (2000) 424-430. | MR 1808183
,[86] Branching of solutions and non-existence of first integrals in Hamiltonian mechanics I, Funct. Anal. Appl. 16 (1982) 181-189. | Zbl 0524.58015
,