Relative equilibria of the four-body problem

Abstract

Abstract. By employing a regularizing transformation, the problem of bifurcation of relative equilibria in the Newtonian 4-body problem is reduced to a study of an algebraic correspondence between real algebraic varieties. The finiteness theorems of algebraic geometry are used to find an upper bound for the number of affine equivalence classes of relative equilibria which holds for all masses in the complement of a proper, algebraic subset of the space of all masses. Introduction; relative equilibria of the N-body problemThe Newtonian TV-body problem concerns the motion of TV point masses under the influence of their mutual gravitational attraction. We will consider the case when all the particles move in a fixed plane which we take to be the complex plane, C. The position vector zeC N is the vector (z,, ...,z n ) where z,-= x,-+ iy,-£ C is the position of the yth particle. The mass vector m e R n+ is (m l ,..., m N ) where m, is the mass of the jth particle. In complex notation, Newton's laws are:

Finiteness of central configurations of five bodies in the plane

“…It is excluded by the condition I 0 > 0. One checks that conditions (16) and (20) corresponding to the remaining two diagrams are impossible with equal positive masses.…”

“…The reader may consult their excellent review on the question. The main works they cite on the subject are Kuz'mina [14], Moeckel [20], Xia [34], Albouy [2], Roberts [26], Moeckel [21]. More recently Hampton [11] proved the finiteness of symmetric planar central configurations of five bodies, except perhaps if the masses satisfy a given polynomial condition.…”

On the number of central configurations in the N-body problem

“…For a modern background one can see (Smale 1970a(Smale , 1970b and (Saari 1980). More recent work can be found in (Buck 1989(Buck , 1991Ced6 and Llibre 1989;Elmabsout 1988;Meyer 1987;Meyer andSchmidt 1988a, 1988b;Moeckel 1985Moeckel , 1989Palmore 1973Palmore , 1975aPalmore , 1975bPacella 1987;Perko and Walter 1985;Schmidt 1988;Shub 1970 andSim6 1977. If ri = (zi, yi, zi) is the position vector of the ith positive mass mi relative to the center of mass of the system, then the particles form a central configuration at time t if and only if there exists some scalar A such that ri = -)~ri for i = 1,2,... ,n. By replacing the acceleration vector rl by the force vector this equation becomes ~t Z r i --rj /~ri = mj r.3.…”

The number of planar central configurations is finite when $N-1$ mass positions are fixed

“…Three of these are collinear and the other two are in the form of an equilateral triangle. For n = 4, a partial answer has been obtained by Moeckel [6]. He proved that the set of mass 4-tuples for which the answer is "no" must have measure zero.…”