# 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

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topDuval, 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 -

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