Top Document: Invariant Galilean Transformations On All Laws Previous Document: 16. But Maxwell's Equations Aren't Galilean Invariant? See reader questions & answers on this topic!  Help others by sharing your knowledge 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')=[ (xvt)(x.cvt) ]=(xx.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, noninvariance 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(xx.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(xvt 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 (xx.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 (xx.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 !?!?!?!?!?!?!?!?!? ! Eleaticus Oren C. Webster ThnkTank@concentric.net ? ! "Anything and everything that requires or encourages systematic ? ! examination of premises, logic, and conclusions" ? !?!?!?!?!?!?!?!?!? User Contributions:Top Document: Invariant Galilean Transformations On All Laws Previous Document: 16. But Maxwell's Equations Aren't Galilean Invariant? Single Page [ Usenet FAQs  Web FAQs  Documents  RFC Index ] Send corrections/additions to the FAQ Maintainer: Thnktank@concentric.net (Eleaticus)
Last Update March 27 2014 @ 02:12 PM

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