Euler configurations and quasi-polynomial systems

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.

21 pages, 6 figures

References in corpus (2)

• Convex Four Body Central Configurations with Some Equal Masses
• Mystery of point charges

Cited by in corpus (7)

• Symmetry of Planar Four-Body Convex Central Configurations
• Convex Four Body Central Configurations with Some Equal Masses
• Saari's Homographic Conjecture of the Three-Body Problem
• On the uniqueness of trapezoidal four-body central configurations
• On the Uniqueness of Co-circular Four Body Central Configurations
• Generic uniqueness of the minimal Moulton central configuration
• Planar $N$-body central configurations with a homogeneous potential