  In this letter we present a theorem on the dynamics of the generalized
Hubbard models. This theorem shows that the symmetry of the single particle
Hamiltonian can protect a kind of dynamical symmetry driven by the
interactions. Here the dynamical symmetry refers to that the time evolution of
certain observables are symmetric between the repulsive and attractive Hubbard
models. We demonstrate our theorem with three different examples in which the
symmetry involves bipartite lattice symmetry, reflection symmetry and
translation symmetry, respectively. Each of these examples relates to one
recent cold atom experiment on the dynamics in the optical lattices where such
a dynamical symmetry is manifested. These experiments include expansion
dynamics of cold atoms, chirality of atomic motion within a synthetic magnetic
field and melting of charge-density-wave order. Therefore, our theorem provides
a unified view of these seemingly disparate phenomena.
