Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom

TL;DR

This paper refines the necessary conditions for integrability of homogeneous Hamiltonian systems with multiple degrees of freedom, providing an algorithm to identify finite eigenvalue sets and discovering new integrable potentials in three-dimensional cases.

Contribution

It sharpens the Morales-Ramis theorem by establishing finite eigenvalue sets for integrability and introduces an algorithm to find these sets, leading to the discovery of new integrable potentials.

Findings

Identified finite eigenvalue sets for integrability conditions.

Discovered new integrable potentials in three degrees of freedom.

Found potentials with higher-degree first integrals.

Abstract

We consider natural complex Hamiltonian systems with $n$ degrees of freedom given by a Hamiltonian function which is a sum of the standard kinetic energy and a homogeneous polynomial potential $V$ of degree $k>2$. The well known Morales-Ramis theorem gives the strongest known necessary conditions for the Liouville integrability of such systems. It states that for each $k$ there exists an explicitly known infinite set $\scM_k\subset\Q$ such that if the system is integrable, then all eigenvalues of the Hessian matrix $V''(\vd)$ calculated at a non-zero $\vd\in\C^n$ satisfying $V'(\vd)=\vd$, belong to $\scM_k$. The aim of this paper is, among others, to sharpen this result. Under certain genericity assumption concerning $V$ we prove the following fact. For each $k$ and $n$ there exists a finite set $\scI_{n,k}\subset\scM_k$ such that if the system is integrable, then all eigenvalues of the Hessian matrix $V''(\vd)$ belong to $\scI_{n,k}$. We give an algorithm which allows to find sets $\scI_{n,k}$. We applied this results for the case $n=k=3$ and we found all integrable potentials satisfying the genericity assumption. Among them several are new and they are integrable in a highly non-trivial way. We found three potentials for which the additional first integrals are of degree 4 and 6 with respect to the momenta.