Kowalevski's analysis of the swinging Atwood's machine

TL;DR

This paper investigates the singularity structure of the swinging Atwood's machine using Kowalevski expansions, revealing numerous mass ratios with maximal arbitrary constants, but it does not determine integrability.

Contribution

It demonstrates the existence of infinite Kowalevski expansions with maximal arbitrary constants for various mass ratios, highlighting the weak Painlevé property of the system.

Findings

Multiple mass ratios admit Kowalevski expansions with maximal arbitrary constants.

Expansions are of weak Painlevé type, indicating complex singularity structures.

Cannot distinguish between integrable and non-integrable cases based solely on these expansions.

Abstract

We study the Kowalevski expansions near singularities of the swinging Atwood's machine. We show that there is a infinite number of mass ratios $M/m$ where such expansions exist with the maximal number of arbitrary constants. These expansions are of the so--called weak Painlev\'e type. However, in view of these expansions, it is not possible to distinguish between integrable and non integrable cases.