Electromagnetic Lagrangian Invariance

[PeroK]

This is an example from "Noether's Theorem" by Neuenschwander. Chapter 5, example 4, page 74-75.

He gives the Lagrangian for a charged particle in an electromagnetic field:

$L=\frac12 m \dot {\vec{r}}^2+e \dot{\vec{r}} \cdot \vec{A} -eV$

And claims invariance invariance under the transformation:

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

And gives $\frac12 \vec{p} \cdot \vec{r} - Ht$ as the conserved quantity, where $\vec{p} = m\dot{\vec{r}} + e\vec{A}$

First, I didn't see any way this was going to work out in general. Then, I considered the electric field associated with a charged particle:

$\vec{A} = 0; \ V = \frac{V_0}{r}$

And, that doesn't lead to the required invariance, as far as I can see.

Any ideas?

 

[PeroK]

I went back to this problem and have got the following equations that must be satisfied for "invariance" of the functional under the above transformation:

$$\vec{\nabla} A_x \cdot \vec{r} + A_x + 2 \frac{\partial A_x}{\partial t} t = 0$$
$$\vec{\nabla} V \cdot \vec{r} + 2V + 2 \frac{\partial V}{\partial t} t = 0$$

With, obviously, the same equation for $A_y$ and $A_z$.

Since there are separate equations for the electric and magnetic potentials, I don't see how they can be satisfied for every EM field. And, in fact, for the simple, static example, the $V$ equation is not satisfied.

I would be interested whether these equations are known in the theory of EM? My provisional assumption at this stage is that Neuenschwander has got this wrong.

Note: the reason I went back to this is that a later problem (6.5) looks at the invariance of the EM Lagrangian under a Lorentz Transformation. Given that the Kinetic Energy is in the wrong form for invariance under Lorentz, this seems to me an even more bizarre and unlikely problem. I already know that the terms due to the KE are not going to cancel out and I don't see how any EM potentials will change that. I might post something on this once I've looked at it.

If anyone has studied this book, perhaps they can enlighten me about what is going on here. I'm concerned I'm missing something fundamental.