Gauge invariance of Euler-Lagrange equations

[andrewkirk]

I have been trying to teach myself Lagrangian mechanics from a textbook “Lagrangian and Hamiltonian Mechanics” by MC Calkin. It has covered virtual displacements, generalised coordinates, d’Alembert’s principle, the definition of the Lagrangian, the Euler-Lagrange differential equation and how it can be used to derive equations of motion for a system.

In discussing the motion of a charged particle in electromagnetic fields, the book introduces the concept of gauge transformations and how they make no change to the derived equations of motion. I can follow the logic of that, but it also claims that it is “easy to show” that the Euler-Lagrange equation $$\frac{d}{dt}(\frac{\partial L}{\partial\.q_a}) = \frac{\partial L}{\partial q_a}$$ is unchanged by the addition to the Lagrangian L of a term that is a total time derivative of a real scalar field $$\lambda(t,x,y,z)$$ defined in the space-time under consideration. I can't see why this should be the case, and I have been unable to turn up a proof through internet searching, although I have found another site that suggests it is true.

Can anyone point me to a place that proves this result?

Thanks very much for any help.