Radiating Systems, Multipole Fields and Radiation — John David Jackson
Physics — E&M · gauge-invariance-lagrangian
Problem
(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 Aα→Aα+∂αΛ of the potentials in the charged-particle Lagrangian (12.12) merely generates another equivalent Lagrangian.