Measure of Degenerate Relative Equilibria. I

Abstract

Relative equilibria of three positive masses always correspond to the well known central configurations found by Euler and Lagrange-the collinear and equilateral triangle solutions of the three body problem. The name equilibrium suggests the fact that each of the masses appears at rest in a rotating barycentric coordinate system. By identifying any two configurations provided that one can be made congruent to the other by a rotation followed by a scalar multiplication, we define a relative equilibria class. Wintner [7] proves that there are only five relative equillibria classes in this case. The three Euler classes and the two Lagrange classes are always nondegenerate [3]. In [2] we showed how a degeneracy arises in the four body problem. In the plane E2 we place three unit masses at the vertices of an equilateral triangle with center of mass at the origin. We place at the origin an arbitrary fourth positive mass, mi. It follows easily for all values of m4 that this configuration is a relative equilibrium. We find that the relative equilibria class to which this configuration belongs is degenerate when m4 equals a unique positive mass ma* < 1.