Some insights from total collapse

TL;DR

This paper explores the historical and theoretical significance of the Sundman-Weierstrass theorem on total collapse in the Newtonian N-body problem, highlighting its role in the shift from quantitative to qualitative analysis of dynamical systems.

Contribution

It provides a historical overview and connects the theorem to key concepts like integrability and singularities, illustrating its impact on the development of dynamical systems theory.

Findings

Highlights the theorem's role as a stability criterion

Connects total collapse to integrability and singularities

Illustrates the shift to qualitative analysis in dynamics

Abstract

We discuss the Sundman-Weierstrass theorem of total collapse in its historical context. This remarkable and relatively simple result, a type of stability criterion, is at the crossroads of some interesting developments in the gravitation Newtonian N-body problem. We use it as motivation to explore the connections to such important concepts as integrability, singularities and tyipicality in order to gain some insight on the transition from a predominantly quantitative to a novel qualitative approach to dynamical problems that took place at the end of the 19th century.