  In this paper we aim at presenting a concise but also comprehensive study of
time-dependent (tdependent) Hamiltonian dynamics on a locally conformal
symplectic (lcs) manifold. We present a generalized geometric theory of
canonical transformations and formulate a time-dependent geometric
Hamilton-Jacobi theory on lcs manifolds. In contrast to previous papers
concerning locally conformal symplectic manifolds, here the introduction of the
time dependency brings out interesting geometric properties, as it is the
introduction of contact geometry in locally symplectic patches. To conclude, we
show examples of the applications of our formalism, in particular, we present
systems of differential equations with time-dependent parameters, which admit
different physical interpretations as we shall point out.
