Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials

TL;DR

This paper extends the understanding of Hamiltonian system integrability by showing that non-diagonalizable Hessian matrices with large Jordan blocks obstruct integrability, using differential Galois theory and Kronecker's ideas.

Contribution

It introduces new obstructions to integrability for Hamiltonian systems with homogeneous potentials when the Hessian matrix is not diagonalizable.

Findings

Jordan blocks of size greater than two prevent Liouville integrability.

Non-diagonalizable Hessians provide additional obstructions beyond eigenvalue conditions.

Differential Galois theory is used to establish these new integrability obstructions.

Abstract

In this paper, we consider the natural complex Hamiltonian systems with homogeneous potential $V(q)$, $q\in\C^n$, of degree $k\in\Z^\star$. 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 $V"(c)$ calculated at a non-zero point $c\in\C^n$, 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.