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Generalized Euler's configurations and Kushnirenko principle
by
Alain Albouy
Institut de Mécanique Céleste, Observatoire de Paris, France
Following A.G. Khovanskii, ``the ideology of the theory of fewnomials'', that we call the Kushnirenko principle, is that ``real varieties defined by simple not cumbersome systems of equations should have a simple topology''. We present a nice result in this direction. We call a [generalized] Euler configuration a relative equilibrium configuration of three particles on a line, with real masses (m1, m2, m3). The interaction generalizes the Newtonian one in the sense that the force is in rb instead of r-2, where b is any real number. The result is the following: given any order of the particles, there exist at most three Euler configurations, except in five cases where any configuration is a solution: (i) m1=m2=m3=0, (ii) b=0 and m1=-m2=m3, (iii) b=1, (iv) b=2, m2=0 and m1=m3, (v) b=3 and m1=m2=m3.
The technics are elementary but the (one page) proof was not easy to find. We worked at it thinking to get a more precise idea about which are the missing mathematical tools that would give an answer to the Chazy-Wintner-Smale conjecture in its simplest case, four masses in the plane.
Date received: June 6, 2003
Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calo-80.