I have been having trouble with a bunch of examples to do with the Rund Trauman Identity. I have also posted this query on Physics help forum but I generally don't have much luck on that forum.

I have the Rund-Trautman identity in this form/notation:

Now for the electromagnitism problem I have the Lagrangian:

I wish to test for invariance under the transformation:

Now the electric scalar potential and the magnetic scalar potential are the bits that are giving me trouble. If I consider just the scalar potential and I use spherical coordinates then I get:

From here I am stuck. Perhaps I need to show that this is equal to the derivative of a function of time, and my calculus just isn't good enough. Or maybe I need to add the terms for the magnetic vector potential and then (somehow) use one of Maxwells equations to show that the result is zero?

In any case I have tried many of these things and am stuck. Can someone suggest an approach?