  Relaxation of few-body quantum systems can strongly depend on the initial
state when the system's semiclassical phase space is mixed, i.e., regions of
chaotic motion coexist with regular islands. In recent years, there has been
much effort to understand the process of thermalization in strongly interacting
quantum systems that often lack an obvious semiclassical limit. Time-dependent
variational principle (TDVP) allows to systematically derive an effective
classical (nonlinear) dynamical system by projecting unitary many-body dynamics
onto a manifold of weakly-entangled variational states. We demonstrate that
such dynamical systems generally possess mixed phase space. When TDVP errors
are small, the mixed phase space leaves a footprint on the exact dynamics of
the quantum model. For example, when the system is initialized in a state
belonging to a stable periodic orbit or the surrounding regular region, it
exhibits persistent many-body quantum revivals. As a proof of principle, we
identify new types of "quantum many-body scars", i.e., initial states that lead
to long-time oscillations in a model of interacting Rydberg atoms in one and
two dimensions. Intriguingly, the initial states that give rise to most robust
revivals are typically entangled states. On the other hand, even when TDVP
errors are large, as in the thermalizing tilted-field Ising model, initializing
the system in a regular region of phase space leads to slowdown of
thermalization. Our work establishes TDVP as a method for identifying
interacting quantum systems with anomalous dynamics in arbitrary dimensions.
Moreover, the mixed-phase space classical variational equations allow to find
slowly-thermalizing initial conditions in interacting models. Our results shed
light on a link between classical and quantum chaos, pointing towards possible
extensions of classical Kolmogorov-Arnold-Moser theorem to quantum systems.
