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.Comment: 21 pages, 6 figure

Saari's homographic conjecture of the three-body problem

“…In the planar three-body problem, the number of central configurations is five. This fact was proved by Moulton [17] for a > −1 and by Albouy [2] for a > −2. Hampton and Moeckel [10] recently showed that the number of central configurations in the Newtonian planar fourbody problem is finite.…”

“…The equations of motion in Fujiwara coordinates for constant configurational measure solutions are given by substituting equations (1), (2), and (3) into (15). We thus obtain…”

“…We will call them the rectilinear central configurations 1, 2, and 3, in agreement with the index of the middle mass, namely (m 3 , m 1 , m 2 ), (m 1 , m 2 , m 3 ), and (m 2 , m 3 , m 1 ), respectively. Each case corresponds to a certain value of the configurational measure; we call them the critical values of µ and denote them by µ (1) c , µ (2) c , and µ…”

“…Since m Q = 0, we have the identity (2,3,1) or (3,1,2), where ∆ is twice the oriented area for the triangle…”

On the central configurations in the spatial 5-body problem with four equal masses

“…In the following we describe the results obtained. Notice that the bifurcation point m f shows a bifurcation inside the set of convex spatial central configurations, while it is conjectured that in the planar 4-body problem there is no bifurcation inside the set of convex planar central configurations, see for instance Albouy and Fu (2007). …”