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Thread: rigid vs. nonrigid transformations

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    Raider of the lost time AntonioLao's Avatar
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    rigid vs. nonrigid transformations

    There are at least three special kinds of rigid transformations. These are, not in any order of importance, translation (T), rotation (R), and reflection (M). Using Cartesian coordinate geometry of 2D Euclidean spacetime, the following mappings always hold true.

    [math]T(x,t)=(x+\\Delta x, t+ \\Delta t)[/math]

    [math]R(x,t)=(xcos\theta-tsin\theta, xsin\theta+tcos\theta)[/math]

    [math]M_S(x,t)=(x, -t)[/math] reflection in the space axis.

    [math]M_T(x,t)=(-x, t)[/math] reflection in the time axis.

    The distinction of all rigid transformations is that they do not stretch (dilate), shrink (contract), or twist (curve) the transformed events. Two transformed events do not move closer together or farther apart in spacetime. In other words, events always stay the same distance from each other. Furthermore, the Pythagorean Theorem determines this distance. The mechanics is Newtonian and the spacetime is Galilean.

    Nonrigid transformations give relativistic mechanics and Minkowskian spacetime. For transformed events involving dilation and contraction refer to Einsteinís special theory of relativity and for the ones involving twisting and curvatures refer to Einsteinís general theory of relativity.

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    The Thinker Guille's Avatar
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    Is there any other "non-main" rigid/non-rigid kind of transformation?

    What about division (breaking)?

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    Raider of the lost time AntonioLao's Avatar
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    Quote Originally Posted by GUILLE
    What about division (breaking)?
    The divisions of vectors are undefined as far as I know. I'm not sure about tensors divisions.

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    Quote Originally Posted by AntonioLao
    The divisions of vectors are undefined as far as I know. I'm not sure about tensors divisions.
    A division of something mathematical as a vector I think that can only lead to other vectors (depending on the number of parts resulting).

    If not, is it possible that, let's say, if a tensor is devided, then the parts are vectors?

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    GUILLE,

    Vectors have two components, the magnitude and the direction. Although magnitude can be divided, direction cannot.

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    Quote Originally Posted by AntonioLao
    GUILLE,

    Vectors have two components, the magnitude and the direction. Although magnitude can be divided, direction cannot.
    Can the parts of the magnitude re-unify?

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    Raider of the lost time AntonioLao's Avatar
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    Quote Originally Posted by GUILLE
    Can the parts of the magnitude re-unify?
    parts re-unified signify continuity and divided signify quantization.

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    Post Re: rigid vs. nonrigid transformations

    Quote Originally Posted by AntonioLao View Post
    parts re-unified signify continuity and divided signify quantization.
    Quite right, what about the process of division, I have been thinking about this fine point for awhile, the process it seems is continuous and the result is defined, but if we can agree to a mathematical process of division, then I think we could define this universally for all particles. It is necesarily discrete and immaterial what process we agree on so long as it is scaleable.

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    Re: rigid vs. nonrigid transformations

    Quote Originally Posted by theunify
    but if we can agree to a mathematical process of division
    Logarithmically, a process of division becomes a process of subtraction where the dividend is replaced by the minuend and the divisor is replaced by the subtrahend.
    Time independence: [∂E(g)]≤=[∂F(a)◊∂r(a)][∂F(b)◊∂r(b)] and Mass independence: a(t)∑r(t)=c

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