STABILITY OF EQUILIBRIA FOR A CLASS OF TIME-REVERSIBLE, D(N)XO(2)-SYMMETRICAL HOMOGENEOUS VECTOR-FIELDS

First-order, time-reversible n-body problems in three-space whose velo city fields consist of sums of identical two-body interactions are stu died under a set of natural symmetry assumptions. Up to linearization about maximally symmetric equilibria, the entire class is shown to be represented by a two-parameter normal form. The symmetries of the clas s are used to find formulas for the eigenvalues of the linearized prob lems. The class of problems is divided into two families, one in which vector field components in the spatial directions act in concert, and one in which they act in opposition. When the components act in conce rt, the equilibria are (i) unstable when interaction strength grows wi th distance, (ii) stable when interaction strength decays and n = 3, 4 , and (iii) stable or unstable when interaction strength decays and n > 4, depending as the singularity of the vector field varies across a critical value. When components act in opposition, stability and insta bility are interchanged. A nonlinear application of this analysis is t he establishment of symmetric near-equilibrium periodic solutions.