  Recent advances in optical Feshbach resonance technique have enabled the
experimental investigation of atomic gases with time-dependent interaction. In
this work, we study the many-body dynamics of weakly interacting bosons subject
with an arbitrary time varying scattering length. By employing a variational
ansatz, we derive an effective Hamiltonian that governs the dynamics of thermal
particles. Crucially, we show that there exists a hidden symmetry in this
Hamiltonian that can map the many-body dynamics to the precession of an SU(1,1)
"spin". As a demonstration, we calculate the situation where the scattering
length is sinusoidally modulated. We show that the non-compactness of the
SU(1,1) group naturally leads to solutions with exponentially growth of
Bogoliubov modes and causes instabilities.
