  We consider a many-body generalization of the Kapitza pendulum: the
periodically-driven sine-Gordon model. We show that this interacting system is
dynamically stable to periodic drives with finite frequency and amplitude. This
finding is in contrast to the common belief that periodically-driven unbounded
interacting systems should always tend to an absorbing infinite-temperature
state. The transition to an unstable absorbing state is described by a change
in the sign of the kinetic term in the effective Floquet Hamiltonian and
controlled by the short-wavelength degrees of freedom. We investigate the
stability phase diagram through an analytic high-frequency expansion, a
self-consistent variational approach, and a numeric semiclassical calculations.
Classical and quantum experiments are proposed to verify the validity of our
results.
