  We analyze quantum dynamics of strongly interacting, kinetically constrained
many-body systems. Motivated by recent experiments demonstrating surprising
long-lived, periodic revivals after quantum quenches in Rydberg atom arrays, we
introduce a manifold of locally entangled spin states, representable by
low-bond dimension matrix product states, and derive equations of motions for
them using the time-dependent variational principle. We find that they feature
isolated, unstable periodic orbits, which capture the recurrences and represent
nonergodic dynamical trajectories. Our results provide a theoretical framework
for understanding quantum dynamics in a class of constrained spin models, which
allow us to examine the recently suggested explanation of 'quantum many-body
scarring' [Nature Physics (2018), doi:10.1038], and establish a connection to
the corresponding phenomenon in chaotic single-particle systems.
