Rund-Trautman Identity Electromagnetic Field

[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?

 

[Vailanor]

To show the invariance of the Lagrangian under the transformation, you can use the chain rule to express the partial derivatives in terms of the primed coordinates. For example, for $V$, you have \begin{equation*}\frac{\partial V}{\partial r'} = \frac{\partial V}{\partial r}\frac{\partial r}{\partial r'} + \frac{\partial V}{\partial t}\frac{\partial t}{\partial r'}.\end{equation*}Now you can plug these into your expression and simplify it using the transformation you gave. You should be able to show that the right-hand side of your equation is equal to 0.