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Finiteness of kite relative equilibria in the five-vortex and five-body problems. (English) Zbl 1195.70020

Summary: We study the finiteness of planar relative equilibria of the Newtonian five-body problem and in the five-vortex problem in the case that configurations form a symmetric kite (three points on a line and two additional points placed symmetrically with respect to that line). We can prove that the equivalence classes of such relative equilibria are finite with some possible exceptional cases. These exceptional cases are given explicitly as polynomials in the masses (or vorticities in the vortex problem). These results depend on computations performed with the software Sage, Singular, Magma, and Gfan.

MSC:

70F10 \(n\)-body problems
70F15 Celestial mechanics
37N05 Dynamical systems in classical and celestial mechanics
14T05 Tropical geometry (MSC2010)
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