Euler configurations and quasi-polynomial systems

Details Euler configurations and quasi-polynomial systems Euler configurations and quasi-polynomial systems

2006-03-29

arxiv.org/abs/math-ph/0603075v1

Physics

Mathematical Physics

21 pages, 6 figures

Scientific paper

10.1134/S1560354707010042

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.

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