Integrability of natural Hamiltonian systems with homogeneous potentials of degree zero

TL;DR

This paper establishes necessary conditions for the integrability of natural Hamiltonian systems with degree-zero homogeneous potentials, linking integrability to properties of the Hessian matrix at specific solutions.

Contribution

It introduces new integrability criteria based on the eigenvalues and structure of the Hessian matrix for such Hamiltonian systems.

Findings

Hessian matrix eigenvalues must be integers for integrability

Hessian matrix must be semi-simple if the system is integrable

Derived conditions apply to systems with degree-zero homogeneous potentials

Abstract

We derive necessary conditions for integrability in the Liouville sense of natural Hamiltonian systems with homogeneous potential of degree zero. We derive these conditions through an analysis of the differential Galois group of variational equations along a particular solution generated by a non-zero solution $\vd\in\C^n$ of nonlinear equations $\grad V(\vd)=\vd$. We proved that if the system integrable then the Hessian matrix $V''(\vd)$ has only integer eigenvalues and is semi-simple.