# 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.

## Key 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.