We can look at infinitesimal transformations in the fields that leaves the Lagrangian invariant, because that implies that the equations of motions are invariant under this transformations. But what really matters is the those transformations that leaves the action invariant. So we can always add a total time derivative of a functions such that the change in the Lagrangian under a infinitesimal transformation is proportional to a total time derivative of a function. My question is, what is the physical interpretation of this? What does this total time derivative of a function represent?