# Rund-Trautman Identity Electromagnetic Field

1. Apr 12, 2013

### Kiwi_Dave

I have been having trouble with a bunch of examples to do with the Rund Trauman Identity.

I have the identity in this form/notation:

$$\frac {\partial L}{\partial q^ \mu}\zeta ^\mu+p_\mu \dot \zeta^\mu+\frac{\partial L}{\partial t}\tau-H \dot \tau=\frac{dF}{dt}$$

Now for the electromagnitism problem I have the Lagrangian:

$$L=\frac12 m \underline{\dot r}^2+q \underline v \cdot \underline A -qV$$

I wish to test for invariance under the transformation:

$$t'=t(1+\epsilon)$$ $$x^{\mu'}=x^{\mu}(1+\frac12 \epsilon)$$

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:

$$\frac {\partial qV}{\partial r}\frac r2+\frac{\partial qV}{\partial t}t=$$

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?