A note on the rational non-integrability of the $N$-center problem for almost all degrees of the singularities

TL;DR

This paper proves that the N-center problem with rational forces is generally not rationally integrable for most singularity orders, except for finitely many specific cases, and outlines conditions for those cases.

Contribution

It establishes non-integrability for almost all rational singularity orders in the N-center problem and characterizes the special cases where integrability may occur.

Findings

Non-integrability for most rational singularity orders

Finite exceptions where integrability is possible

Necessary conditions for integrability in remaining cases

Abstract

In this article, we show that the $N$-center problem with rational weak and moderate forces is not rationally integrable for all but a finite number of values $\alpha\in(0,2)\cap \mathbb{Q}$, where $\alpha$ is the order of the singularities. We identify the remaining cases and provide the necessary conditions for their integrability.