Re: field transformations

[Jack Uretsky <jlu@HEP.ANL.GOV>]

        I've seen this done (possibly by Eyges).  You can start with
Newton's Laws and note that they are invariant under shifts in the
velocity of a frame.  Then note that Maxwell's equations are not, so the
question is: What's the transformation that prseserves Maxwell's
equations?  With some simple restrictions, like linearity, you end up with
the Lorentz equations.

On Wed, 2 Apr 2003, Joe Heafner wrote:

> On Friday, Mar 28, 2003, at 18:06 US/Eastern, John S. Denker wrote:
>
> > I can't imagine deriving or even explaining the
> > transformations without four-vectors.  For that
> > matter I can't imagine introductory E&M without
> > four-vectors.
>
> Well, I meant "introductory" in the sense of the second semester of
> that first introductory course (e.g. Halliday and Resnick). The text
> I'm using, Matter & Interactions, presents a very nice discussion of
> reference frames and fields. The authors specifically demonstrate how a
> static E field in one inertial frame is seen as a combined E and B
> field in another inertial frame. They do so using the Lorentz
> transformation of E and B, but they do not present an actual derivation
> of the transformation. They don't present a derivation of the standard
> Lorentz transformation either.
>
> What I've been looking for is a way to derive this transformation
> without using tensors or four vectors. Today, I found a way on pp.
> 231-232 of Eyges' text _The Classical Electromagnetic Field_ (Dover,
> 1980). The method involves Lorentz transforming the components of \/ x
> E = -(1/c)dB/dt and imposing form invariance. I'm in the process of
> carrying out the whole derivation to make sure I understand it.
>
> I agree what geometric methods are probably the most elegant, but I
> don't understand them. I'm just now beginning to think I understand
> tensors. Technically, I "learned" them in grad school from Goldstein
> and Jackson, but in reality what I "learned" were mechanical
> mathematical steps for working problems. The breakthrough for me
> consisted of a thorough understanding of four vectors and a realization
> that scalars, vectors, and four vectors are merely special cases of
> tensors.
>
> Cheers,
> Joe Heafner

--
        "What did Barrow's lectures contain?  Bourbaki writes with some
scorn that in his book in a hundred pages of the text there are about 180
drawings.  (Concerning Bourbaki's books it can be said that in a thousand
pages there is not one drawing, and it is not at all clear which is
worse.)"
                                        V. I. Arnol'd in
                                Huygens & Barrow, Newton & Hooke