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In an ideal accelerator, the single-particle dynamics can be decoupled intotransverse motion -- the betatron oscillations -- and longitudinal motion --the synchrotron oscillations. Chromatic and dispersive effects introduce acoupling between these dynamics, the so-called synchro-betatron coupling. Wepresent an analysis of the fully coupled dynamics over a single synchrotronoscillation that leads to an averaged invariant with synchro-betatron couplingin a generic lattice. We apply this analysis to two problems: first, a toylattice where the computations are analytically tractable, then a design for arapid cycling synchrotron built using the integrable optics described byDanilov and Nagaitsev, showing that although there is fairly complex behaviorover the course of a synchrotron oscillation, the Danilov-Nagaitsev invariantsare nevertheless periodic with the synchrotron motion.

Averaged Invariants in Storage Rings with Synchrotron Motion