Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials

Duval, Guillaume; Maciejewski, Andrzej J.

Annales de l'Institut Fourier, Tome 59 (2009) no. 7, pp. 2839-2890.

Dans cet article, nous étudions les systèmes Hamiltoniens de potentiels homogènes $V\left(q\right)$, $q\in {ℂ}^n$ de degré $k\in {\mathbb{Z}}^{*}$. Morales et Ramis avaient donné des conditions nécessaires à l’intégrabilité de ces sytèmes en termes des valeurs propres des matrices de Hessienne $V^{\prime \prime }\left(c\right)$, calculées aux points $c\in {ℂ}^n$ tels que $V^{\prime }\left(c\right)=c$. Le thème principal de ce travail est de montrer que d’autres obstructions à l’intégrabilité apparaissent quand $V^{\prime \prime }\left(c\right)$ n’est pas diagonalisable. Entre autres, nous prouvons que si $V^{\prime \prime }\left(c\right)$ 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.

In this paper, we consider the natural complex Hamiltonian systems with homogeneous potential $V\left(q\right)$, $q\in {ℂ}^n$, of degree $k\in {\mathbb{Z}}^{☆}$. 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 $V^{\prime \prime }\left(c\right)$ calculated at a non-zero point $c\in {ℂ}^n$, such that $V^{\prime }\left(c\right)=c$. The main aim of this paper is to show that there are other obstructions for the integrability which appear if the matrix $V^{\prime \prime }\left(c\right)$ is not diagonalizable. We prove, among other things, that if $V^{\prime \prime }\left(c\right)$ 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.

Classification : 37J30, 70H07, 37J35, 34M35
Keywords: Hamiltonian systems, integrability, differential Galois theory
Mot clés : systèmes hamiltoniens, intégrabilité, théorie de Galois différentielle

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