  Cosymplectic geometry has been proven to be a very useful geometric
background to describe time-dependent Hamiltonian dynamics. In this work, we
address the globalization problem of locally cosymplectic Hamiltonian dynamics
that failed to be globally defined. We investigate both the geometry of locally
conformally cosymplectic (abbreviated as LCC) manifolds and the Hamiltonian
dynamics constructed on such LCC manifolds. Further, we provide a geometric
Hamilton-Jacobi theory on this geometric framework.
