Doesthetransiti~nt~cha~~ Determinethedynamicaperture?*

We review the important notion of the dynamic aperture of a storage ring with emphasis on its relation to general ideas of dynamical instability, notably the transition to chaos. Practical approaches to the problem are compared. We suggest a somewhat novel quantitative guide to the old problem of choosing machine tunes based on a heuristic blend of KAM theory and resonance selection rules. INTRODUCTION Nowadays, machine designers are much exercised by the dynamic apertures of their storage rings. Roughly speaking, the dynamic aperture, or non-linear acceptance, is the region around the central closed orbit in which single particle motion is stable. The term itself only recently came into use but neatly encapsulates the essence of what it is meant to describe. Formerly, the aperture tout court of a synchrotron or storage ring was what is now variously described as the physical or mechanical acceptance or aperture ; that is, the aperture determined by the vacuum chamber or other material obstruction presented to the beam. To explain the distinction, we recall that, in terms of action-angle variables of linearised motion (R.