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Invariant Galilean Transformations On All Laws
Section - 17. First and Second Derivative differential equations.

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One of the intellectually corrupt ways of
denying the very simple demonstration of
galilean invariance of all laws expressed
in the generalized coordinate form demanded
by analytic geometry, vector analysis, and
measurement theory 

    [ (x'-x.c')=[ (x-vt)-(x.c-vt) ]=(x-x.c) ]

is the assertion that those equations 'over there'
(usually Maxwell or wave) are somehow immune to
the elementary laws of algebra used to demon-
strate the invariance.  [Unfortunately, the 
assertions are never accompanied by reference
to the magical math that makes elementary al-
gebra invalid.  Wonder why that is?]

Part of the time it is based on the old lore
based on the incompetent transformation of
the privileged form of an equation instead
of the correct form. [Evidence of this is
any reference to an effect due to the velocity
of the transform; it falls out algebraicly
- as you see above - and cancels out arith-
metically - as you can see above.]

But usually it is just whistling in the dark, 
waving the cross (zwastika, I'd say) at 
the mean old vampire.

The most general equation that could be conjured 
up is a differential with either First or Second
Derivatives.

Let's examine the plausibility of such magical
magical, non-invariance assertions.

(a) to get a Second Derivative you must have
    a First Derivative.
(b) to get a First Derivative you must have
    a function to differentiate.
(c) to get a Second Derivative you must have
    a function in the second degree.

So, let us examine the question as to whether
any such common Maxwell/wave equation will 
differ for 

(a) the common, privileged form, represented
    as ax^2, with a being an unknown constant
    function.

(b) the generalized cartesian form, represented
    as a(x-x.c)^2 = ax^2 -2ax(x.c) + ax.c^2,
    with a being an unknown constant function.

(c) the transformed generalized cartesian form, 
    represented as a(x-vt -x.c+vt)^2, same as for 
    (b), = ax^2 -2ax(x.c) + ax.c^2, of course, 
    with a being an unknown constant function.

I.  for (a), remembering that x.c is a constant,
    and that this version is only correct because
    x.c=0, otherwise (b) is the correct form:
     
     d/dx    ax^2  = 2ax
    (d/dx)^2 ax^2  = 2a


II. for (b), remembering that x.c is a constant.

     d/dx    (ax^2 -2ax(x.c) + ax.c^2) = 2ax - 2ax.c
    (d/dx)^2 (ax^2 -2ax(x.c) + ax.c^2) = 2a


III. for (c); same as for (b).


So, what we have seen so far is 

(1)  differential equations in the second degree 
- the wave equations - must clearly be the same for
all forms: the privileged form in x, the generalized
cartesian form in x and the centroid, x.c, or the
transformed generalized cartesian form.

That is, anyone who imagines that correct usage 
gives different results for galilean transformed
frames is at first showing his ignorance, and in
the end showing his intellectual corruption.

(2) As far as the First Derivatives are concerned, the
only cases in which there really is a difference between
the two forms is where x.c <> 0, and in that case, the
use of the privileged form is obviously incompetent.

So, how do you correctly use the differential equations?

If you are using rest frame data with the centroid
at x=0, etc, you can't go wrong without trying to 
go wrong.

If you are using rest frame data with the centroid
not at x=0, you must use (x-x.c) anyplace x appears
in the equation.

If you are using moving frame data, you must use the
moving frame centroid as well as the light front
(or whatever) moving frame data itself, perhaps first 
calculating (x'-x.c'), which equals (x-x.c) which is 
obviously correct, and which is obviously the plain old 
correct x of the privileged form.

Unless, of course, there really is some magical term
or expression that invalidates the obvious and elemen-
tary algebra of the invariance demonstration.

Or maybe you just whistle when you don't want basic
algebra to hold true.



Eleaticus

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! Eleaticus        Oren C. Webster         ThnkTank@concentric.net  ?
! "Anything and everything that requires or encourages systematic   ?
!  examination of premises, logic, and conclusions"                 ?
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Last Update March 27 2014 @ 02:12 PM