JORDAN OBSTRUCTION TO THE INTEGRABILITY OF HAMILTONIAN SYSTEMS WITH HOMOGENEOUS POTENTIALS

Guillaume DUVAL; Andrzej J. MACIEJEWSKI

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JORDAN OBSTRUCTION TO THE INTEGRABILITY OF HAMILTONIAN SYSTEMS WITH HOMOGENEOUS POTENTIALS

机译：JORDAN OBSTRUCTION TO THE INTEGRABILITY OF HAMILTONIAN SYSTEMS WITH HOMOGENEOUS POTENTIALS

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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 Jordanblock of size greater than two, then the system is not integrable in the Lionville 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.

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《Annales de l'Institut Fourier》 |2009年第7期|2839-2890|共52页
作者
Guillaume DUVAL; Andrzej J. MACIEJEWSKI;
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作者单位

Chemin du Chateau 76430 Les Trois Pierres (France);

University of Zielona Gera Institute of Astronomy Podgorna 50 65-246 Zielona Gera (Poland);

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正文语种英语
中图分类自然科学总论;
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Hamiltonian systems; integrability; differential Galois theory;

JORDAN OBSTRUCTION TO THE INTEGRABILITY OF HAMILTONIAN SYSTEMS WITH HOMOGENEOUS POTENTIALS

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