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 Poincaré 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 Poincaré disk. Further considering quantum gases in an oscillating lattice, we also study the dynamics of the system with time-dependent single-particle dispersion.