On the dynamical equivalence of Lagrangians differing up to the total time derivative of an arbitrary function of coordinates and time - A note on the foundation of electromagnetic theory as a gauge theory

It is a remarkable occurence that if two Lagrangians differ up to the total time derivative of an arbitrary function of coordinates and time they are dynamically equivalent, i.e., they imply the same equations of motion despite these Euler-Lagrange equations are derived by a variational procedure or otherwise ("direct problem" versus "inverse problem"). In this note it is shown that by a proper identification of such redundance term the whole of the Maxwell equations can be readily derived, the gauge symmetry of the electromagnetic theory emerging from the outset as the fundamental symmetry.