NUMERICAL INVESTIGATION OF PERIODIC MOTION IN THE THREE-DIMENSIONAL RING PROBLEM OF N BODIES

Abstract

We study numerically the three-dimensional periodic motion of a massless particle in an annular configuration of N bodies. This ring model can approximate the dynamics of celestial systems like clusters and planetary rings. After calculating numerically the families of x-symmetric periodic orbits in the planar case for a certain N, we investigate the families' vertical stability and locate those members that present critical instability in the eigenvalues corresponding to the z-axis. These orbits are the points of bifurcation of the planar families of periodic orbits to three-dimensional families. By means of differentially correcting the vertically critical orbits we extend our numerical investigation in the three-dimensional space and locate several families of three-dimensional periodic orbits symmetric to the xz-plane. The evolution of the families and the significant features of their members are subsequently presented.