  The study of dynamics in closed quantum systems has recently been revitalized
by the emergence of experimental systems that are well-isolated from their
environment. In this paper, we consider the closed-system dynamics of an
archetypal model: spins near a second order quantum critical point, which are
traditionally described by the Kibble-Zurek mechanism. Imbuing the driving
field with Newtonian dynamics, we find that the full closed system exhibits a
robust new phenomenon -- dynamic critical trapping -- in which the system is
self-trapped near the critical point due to efficient absorption of field
kinetic energy by heating the quantum spins. We quantify limits in which this
phenomenon can be observed and generalize these results by developing a
Kibble-Zurek scaling theory that incorporates the dynamic field. Our findings
can potentially be interesting in the context of early universe physics, where
the role of the driving field is played by the inflaton or a modulus.
