Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials
Annales de l'Institut Fourier, Volume 59 (2009) no. 7, p. 2839-2890
In this paper, we consider the natural complex Hamiltonian systems with homogeneous potential V(q), q n , of degree k . The known results of Morales and Ramis give necessary conditions for the complete integrability of such systems. These conditions are expressed in terms of the eigenvalues of the Hessian matrix V (c) calculated at a non-zero point c n , such that V (c)=c. The main aim of this paper is to show that there are other obstructions for the integrability which appear if the matrix V (c) is not diagonalizable. We prove, among other things, that if V (c) contains a Jordan block of size greater than two, then the system is not integrable in the Liouville sense. The main ingredient in the proof of this result consists in translating some ideas of Kronecker about Abelian extensions of number fields into the framework of differential Galois theory.
Dans cet article, nous étudions les systèmes Hamiltoniens de potentiels homogènes V(q), q n de degré k * . Morales et Ramis avaient donné des conditions nécessaires à l’intégrabilité de ces sytèmes en termes des valeurs propres des matrices de Hessienne V (c), calculées aux points c n tels que V (c)=c. Le thème principal de ce travail est de montrer que d’autres obstructions à l’intégrabilité apparaissent quand V (c) n’est pas diagonalisable. Entre autres, nous prouvons que si V (c) possède un bloc de Jordan de taille supérieure à deux, alors le sytème n’est pas intégrable. Pour ce faire, nous avons adapté des idées de Kronecker sur les extensions Abeliennes de corps de nombres, dans le contexte de la théorie de Galois différentielle.
DOI : https://doi.org/10.5802/aif.2510
Classification:  37J30,  70H07,  37J35,  34M35
Keywords: Hamiltonian systems, integrability, differential Galois theory
@article{AIF_2009__59_7_2839_0,
     author = {Duval, Guillaume and Maciejewski, Andrzej J.},
     title = {Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {7},
     year = {2009},
     pages = {2839-2890},
     doi = {10.5802/aif.2510},
     zbl = {1196.37096},
     mrnumber = {2649341},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2009__59_7_2839_0}
}
Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials. Annales de l'Institut Fourier, Volume 59 (2009) no. 7, pp. 2839-2890. doi : 10.5802/aif.2510. https://aif.centre-mersenne.org/item/AIF_2009__59_7_2839_0/

[1] Arnold, V. I. Mathematical methods of classical mechanics, Springer-Verlag, New York, second edition, Graduate Texts in Mathematics, Tome 60 (1989) (Translated from the Russian by K. Vogtmann and A. Weinstein) | MR 997295 | Zbl 0386.70001

[2] Baider, A.; Churchill, R. C.; Rod, D. L.; Singer, M. F. On the Infinitesimal Geometry of Integrable Systems, Mechanics Day (Waterloo, ON, 1992), Fields Inst. Commun., Tome 7 (1996) (Providence, RI: Amer. Math. Soc., pp. 5–56) | MR 1365771 | Zbl 1005.37510

[3] Churchill, R. C Two generator subgroups of SL(2,C) and the hypergeometric, Riemann, and Lamé equations, J. Symbolic Comput., Tome 28 (1999) no. 4-5, pp. 521-545 | Article | MR 1731936 | Zbl 0958.34074

[4] Iwasaki, K.; Kimura, H.; Shimomura, S.; Yoshida, M. From Gauss to Painlevé, Braunschweig: Friedr. Vieweg & Sohn, Aspects of Mathematics, E16 (1991) | MR 1118604 | Zbl 0743.34014

[5] Kimura, T. On Riemann’s Equations Which Are Solvable by Quadratures, Funkcial. Ekvac., Tome 12 (1969/1970), pp. 269-281 | MR 277789 | Zbl 0198.11601

[6] Kolchin, E. R. Algebraic groups and algebraic dependence, Amer. J. Math., Tome 90 (1968), pp. 1151-1164 | Article | MR 240106 | Zbl 0169.36701

[7] Kovacic, J. J. An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Comput., Tome 2 (1986) no. 1, pp. 3-43 | Article | MR 839134 | Zbl 0603.68035

[8] Kozlov, V. V. Symmetries, Topology and Resonances in Hamiltonian Mechanics, Springer-Verlag, Berlin (1996) | MR 1411677 | Zbl 0843.58068

[9] Morales Ruiz, J. J. Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Birkhäuser Verlag, Basel, Progress in Mathematics, Tome 179 (1999) | MR 1713573 | Zbl 0934.12003

[10] Morales Ruiz, J. J.; Ramis, J. P. A note on the non-integrability of some Hamiltonian systems with a homogeneous potential, Methods Appl. Anal., Tome 8 (2001) no. 1, pp. 113-120 | MR 1867496 | Zbl 1140.37353

[11] Morales Ruiz, J. J.; Ramis, J. P. Galoisian obstructions to integrability of Hamiltonian systems. I, Methods Appl. Anal., Tome 8 (2001) no. 1, pp. 33-95 | MR 1867495 | Zbl 1140.37352

[12] Poole, E. G. C. Introduction to the theory of linear differential equations, Dover Publications Inc., New York (1960) | MR 111886 | Zbl 0090.30202

[13] Ramis, E.; Deschamps, C.; Odoux, J. Cours de mathématiques spéciales, Masson, Paris, Algèbre et applications à la géométrie, Tome 2 (1979) | MR 557042 | Zbl 0471.00001

[14] Singer, M.; Van Der Put, M. Galois Theory of Linear Differential Equations, Springer-Verlag, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 328 (2003) | MR 1960772 | Zbl 1036.12008

[15] Yoshida, H. A criterion for the nonexistence of an additional integral in Hamiltonian systems with a homogeneous potential, Phys. D, Tome 29 (1987) no. 1-2, pp. 128-142 | Article | MR 923886 | Zbl 0659.70012