Gravity in one dimension: Stability of periodic orbits

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Abstract
The failure of the one-dimensional gravitational system to relax to equilibrium on predicted time scales has raised questions concerning the ergodic properties of the dynamics. A failure to approach equilibrium could be caused by the segmentation of phase space into isolated regions from which the system cannot escape. In general, each region may have distinct ergodic properties. By numerically investigating the stability of two classes of periodic orbits for the N-body system, we have unequivocally demonstrated that stable regions in the phase space exist for N≤10. For populations 11≤N≤20 we find numerical evidence for multiple, chaotic, invariant regions. Thus the failure of large systems (say, N≥100) to equilibrate may be a result of microscopic dynamical restrictions, rather than imposed macroscopic contraints.