(a) Show from Hamilton's principle that Lagrangians that differ only by a total time derivative of some function of the coordinates and time are equivalent in the sense that they yield the same Euler-Lagrange equations of motion.

(b) Show explicitly that the gauge transformation A^α → A^α + ∂^αA of the potentials in the charged-particle Lagrangian (12.12) merely generates another equivalent Lagrangian.