Stroboscopic Invariants in Storage Rings with Synchrotron Motion

In an ideal accelerator, the single-particle dynamics can be decoupled into transverse motion -- the betatron oscillations -- and longitudinal motion -- the synchrotron oscillations. Chromatic and dispersive effects introduce a coupling between these dynamics, the so-called synchro-betatron coupling. We present an analysis of the fully coupled dynamics over a single synchrotron oscillation that leads to a stroboscopic invariant with synchro-betatron coupling in a generic lattice. This invariant is correct to $\mathcal{O}(\nu_s)$, where $\nu_s$ is the synchrotron tune. We apply this analysis to a design for a rapid cycling synchrotron built using the integrable optics described by Danilov and Nagaitsev, showing that although there is fairly complex behavior over the course of a synchrotron oscillation, the predicted invariants are nevertheless periodic with the synchrotron motion.