Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials

Guillaume Duval[1]; Andrzej J. Maciejewski[2]

  • [1] 1 Chemin du Chateau 76430 Les Trois Pierres (France)
  • [2] University of Zielona Góra Institute of Astronomy Podgórna 50 65–246 Zielona Góra (Poland)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 7, page 2839-2890
  • ISSN: 0373-0956

Abstract

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In this paper, we consider the natural complex Hamiltonian systems with homogeneous potential V ( q ) , q ∈ ℂ n , of degree k ∈ ℤ ☆ . 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 ∈ 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.

How to cite

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Duval, Guillaume, and Maciejewski, Andrzej J.. "Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials." Annales de l’institut Fourier 59.7 (2009): 2839-2890. <http://eudml.org/doc/10474>.

@article{Duval2009,
abstract = {In this paper, we consider the natural complex Hamiltonian systems with homogeneous potential $V(q)$, $q\in \mathbb\{C\}^n$, of degree $k\in \mathbb\{Z\}^\star $. 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 $V^\{\prime\prime\}(c)$ calculated at a non-zero point $c\in \mathbb\{C\}^n$, such that $V^\{\prime\}(c)=c$. The main aim of this paper is to show that there are other obstructions for the integrability which appear if the matrix $V^\{\prime\prime\}(c)$ is not diagonalizable. We prove, among other things, that if $V^\{\prime\prime\}(c)$ 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.},
affiliation = {1 Chemin du Chateau 76430 Les Trois Pierres (France); University of Zielona Góra Institute of Astronomy Podgórna 50 65–246 Zielona Góra (Poland)},
author = {Duval, Guillaume, Maciejewski, Andrzej J.},
journal = {Annales de l’institut Fourier},
keywords = {Hamiltonian systems; integrability; differential Galois theory},
language = {eng},
number = {7},
pages = {2839-2890},
publisher = {Association des Annales de l’institut Fourier},
title = {Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials},
url = {http://eudml.org/doc/10474},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Duval, Guillaume
AU - Maciejewski, Andrzej J.
TI - Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 7
SP - 2839
EP - 2890
AB - In this paper, we consider the natural complex Hamiltonian systems with homogeneous potential $V(q)$, $q\in \mathbb{C}^n$, of degree $k\in \mathbb{Z}^\star $. 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 $V^{\prime\prime}(c)$ calculated at a non-zero point $c\in \mathbb{C}^n$, such that $V^{\prime}(c)=c$. The main aim of this paper is to show that there are other obstructions for the integrability which appear if the matrix $V^{\prime\prime}(c)$ is not diagonalizable. We prove, among other things, that if $V^{\prime\prime}(c)$ 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.
LA - eng
KW - Hamiltonian systems; integrability; differential Galois theory
UR - http://eudml.org/doc/10474
ER -

References

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  1. V. I. Arnold, Mathematical methods of classical mechanics, 60 (1989), Springer-Verlag, New York, second edition Zbl0386.70001MR997295
  2. A. Baider, R. C. Churchill, D. L. Rod, M. F. Singer, On the Infinitesimal Geometry of Integrable Systems, 7 (1996), Mechanics Day (Waterloo, ON, 1992) Zbl1005.37510MR1365771
  3. R. C Churchill, Two generator subgroups of SL ( 2 , C ) and the hypergeometric, Riemann, and Lamé equations, J. Symbolic Comput. 28 (1999), 521-545 Zbl0958.34074MR1731936
  4. K. Iwasaki, H. Kimura, S. Shimomura, M. Yoshida, From Gauss to Painlevé, (1991), Braunschweig: Friedr. Vieweg & Sohn Zbl0743.34014MR1118604
  5. T. Kimura, On Riemann’s Equations Which Are Solvable by Quadratures, Funkcial. Ekvac. 12 (1969/1970), 269-281 Zbl0198.11601MR277789
  6. E. R. Kolchin, Algebraic groups and algebraic dependence, Amer. J. Math. 90 (1968), 1151-1164 Zbl0169.36701MR240106
  7. J. J. Kovacic, An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Comput. 2 (1986), 3-43 Zbl0603.68035MR839134
  8. V. V. Kozlov, Symmetries, Topology and Resonances in Hamiltonian Mechanics, (1996), Springer-Verlag, Berlin Zbl0843.58068MR1411677
  9. J. J. Morales Ruiz, Differential Galois Theory and Non-Integrability of Hamiltonian Systems, 179 (1999), Birkhäuser Verlag, Basel Zbl0934.12003MR1713573
  10. J. J. Morales Ruiz, J. P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential, Methods Appl. Anal. 8 (2001), 113-120 Zbl1140.37353MR1867496
  11. J. J. Morales Ruiz, J. P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems. I, Methods Appl. Anal. 8 (2001), 33-95 Zbl1140.37352MR1867495
  12. E. G. C. Poole, Introduction to the theory of linear differential equations, (1960), Dover Publications Inc., New York Zbl0090.30202MR111886
  13. E. Ramis, C. Deschamps, J. Odoux, Cours de mathématiques spéciales, 2 (1979), Masson, Paris Zbl0471.00001MR557042
  14. M. Singer, M. Van der Put, Galois Theory of Linear Differential Equations, 328 (2003), Springer-Verlag Zbl1036.12008MR1960772
  15. H. Yoshida, A criterion for the nonexistence of an additional integral in Hamiltonian systems with a homogeneous potential, Phys. D 29 (1987), 128-142 Zbl0659.70012MR923886

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