  Motivated by recent advances in quantum dynamics, we investigate the dynamics
of the system with $SU(1,1)$ symmetry. Instead of performing the time-ordered
integral for the evolution operator of the time-dependent Hamiltonian, we show
that the time evolution operator can be expressed as an $SU(1,1)$ group
element. Since the $SU(1,1)$ group describes the "rotation" on a hyperbolic
surface, the dynamics can be visualized on a Poincar\'e disk, a stereographic
projection of the upper hyperboloid. As an example, we present the trajectory
of the revival of Bose-Einstein condensation and that of the scale-invariant
Fermi gas on the Poincar\'e disk. Further considering the quantum gas in the
oscillating lattice, we also study the dynamics of the system with
time-dependent single-particle dispersion. Our results are hopefully to be
checked in current experiments.
