Euler configurations and quasi-polynomial systems

TL;DR

This paper demonstrates that the maximum of three Euler configurations or collinear relative equilibria is a general property of quasi-polynomial systems, unifying different classical problems in celestial mechanics and vortex dynamics.

Contribution

It introduces a quasi-polynomial framework that proves the 'at most three' configurations result in both the 3-body problem and vortex dynamics, providing a unified approach.

Findings

The maximum number of Euler configurations is three for any masses.

The maximum number of collinear relative equilibria is three for point vortices.

The quasi-polynomial system approach applies to both celestial and fluid dynamics problems.

Abstract

In the Newtonian 3-body problem, for any choice of the three masses, there are exactly three Euler configurations (also known as the three Euler points). In Helmholtz' problem of 3 point vortices in the plane, there are at most three collinear relative equilibria. The "at most three" part is common to both statements, but the respective arguments for it are usually so different that one could think of a casual coincidence. By proving a statement on a quasi-polynomial system, we show that the "at most three" holds in a general context which includes both cases. We indicate some hard conjectures about the configurations of relative equilibrium and suggest they could be attacked within the quasi-polynomial framework.