Integrability of Hamiltonian systems with homogeneous potentials of degrees ±2. An application of higher order variational equations

Abstract

The present work is the first one of two papers, in which we analyse systems of higher order variational equations associated to natural Hamiltonian systems with homogeneous potential of degree k ε ℤ\-1; 0; 1}. Our attempt is to give necessary conditions for complete integrability which can be deduced in a framework of differential Galois theory. We show that the higher variational equations VEp of order p ≥ 2, although complicated, have a very particular algebraic structure. More precisely, we show that if VE1 has virtually Abelian differential Galois group (DGG), then VEp are solvable for an arbitrary p > 1. We proved this inductively using what we call the second level integrals. Then we formulate the necessary and sufficient conditions in terms of these second level integrals for VEp to be virtually Abelian. We apply the above conditions to potentials of degree k = ±2 considering their VEp with p > 1 along Darboux points. For k = 2, VE1 does not give any obstruction to the integrability. We show that under certain non-resonance condition, the only degree two integrable potential is the multidimensional harmonic oscillator. In contrast, for degree k = -2 potentials, all the VEp along Darboux points are virtually Abelian.