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

TL;DR

This paper investigates the algebraic structure of higher variational equations in Hamiltonian systems with homogeneous potentials of degrees ±2, establishing conditions for their integrability and identifying the harmonic oscillator as the unique integrable degree 2 potential.

Contribution

It provides a detailed analysis of higher variational equations' structure and criteria for their integrability, specifically applying to potentials of degrees ±2.

Findings

Higher variational equations are solvable if the first variational equation is virtually Abelian.

The harmonic oscillator is the only integrable potential of degree 2 under non-resonance conditions.

All degree -2 potentials have virtually Abelian higher variational equations along Darboux points.

Abstract

The present work is the first of a serie of two papers, in which we analyse the higher variational equations associated to natural Hamiltonian systems, in their attempt to give Galois obstruction to their integrability. We show that the higher variational equations $VE_{p}$ for $p\geq 2$, although complicated they are, have very particular algebraic structure. Preceisely they are solvable if $VE_{1}$ is virtually Abelian since they are solvable inductively by what we call the \emph{second level integrals}. We then give necessary and sufficient conditions in terms of these second level integrals for $VE_{p}$ to be virtually Abelian (see Theorem 3.1). Then, we apply the above to potentials of degree $k=\pm 2$ by considering their $VE_{p}$ along Darboux points. And this because their $VE_{1}$ does not give any obstruction to the integrablity. In Theorem 1.2, we show that under non-resonance conditions, the only degree two integrable potential is the \emph{harmonic oscillator}. In contrast for degree -2 potentials, all the $VE_{p}$ along Darboux points are virtually Abelian (see Theorem 1.3)