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Why does higher derivatives of displacement not concentrated?

  1. Jun 10, 2012 #1
    I luckily went on to Newton's laws which I already know. I was stuck with a fantastic question ( I have been seeing the newton's laws from my childhood, but I didn't notice these sort of observation ) . I was seeing the Force and Momentum. Force is based upon acceleration and Momentum on Velocity. Then I started realizing that they both are derivatives and integrals of each other. But I went further thinking something else. Why cant there be some other higher term which is the derivative of acceleration. What is rate of change of acceleration ?. What is its rate of change ?. What is its rate of Change?....., Why can't we proceed to infinite applying the derivative recursively ?.

    * Why did we stop only at Acceleration which is the 2-nd derivative of displacement ?.

    *That idea has lead to a profound realization of discovering a new space that looks elegant for me. The space is called D-Space. For a D(X)-Space the elements in that space are nothing but functions in X.

    Description of Space : D(X).

    D(X) contains all the functions in X. Every entity " M " in D(X) can be represented by a tuple < d^0(M), d^1(M),d^2(M)........d^n(M).........>. Where d^n(X) is the n-th derivative of X . The minimum 'n' for which the M vanishes is termed by me as Tolerance of M which is an integer ( Tolerance of M represents the patience of M to bear the rude derivatives without vanishing, like we bear the petrol hikes without stopping the usage of vehicles ;) ) .
    So for example SinX belongs to D(X) . SinX = < SinX, CosX, -SinX, -CosX,......> ( Since d^0(SinX)= SinX, d^1(SinX)= CosX....) . Its Tolerance is infinite .

    * If we consider all such tuples we can even add them defining a group operation. For example x^2=<x^2,2x,2,0,0,.....> and x=<x,1,0,0,0...>. Then x^2+x = < x^2+x, 2x+1, 2, 0, 0...>. So the pair wise addition luckily seems to hold perfectly. Even Multiplication too. So we can define a group D(X) with these properties.

    * In the same way there can be I(X) ( Integral group, where every element can be written as the tuple containing derivatives replaced by integrals ). I(X) and D(X) are inverses of each other.

    * So if we are given an D(X) the Displacement tuple contains Infinite tolerance. Displacement is M=< M, Velocity, Acceleration, ......>.

    * So this type of spaces can be used in differential algebra and also geometric analysis. Because we consider some smooth functions everywhere. Those can be easily represented by a map from D(X) .

    That is my idea. I am struggling to find an elegant application and put this forward.
     
  2. jcsd
  3. Jun 10, 2012 #2
    I can answer part of this, at least. Look up the term "jerk." (not the insulting version, the physics version. Just had to put that in there). Basically, it's acclerations first derivative with respect to time.
     
  4. Jun 10, 2012 #3
    But Why cant the higher derivatives still not considered ?
     
  5. Jun 10, 2012 #4
    But Why cant the higher derivatives still not considered ?
     
  6. Jun 10, 2012 #5
    A very fuzzy question, because I'm not sure what are you suggesting. A few notes:

    We didn't stop. When you say it like that it seems like you are implying that Newton's law is result of a science community consensus to obey the Great Master, rather than meticulously verified principle of nature.

    That space is called Fréchet space of infinitely differentiable functions, I believe.

    The fact that you can add them like that is consequence of derivation being linear operator. You probably can define multiplication, as in scalar product of vectors, but I'm not sure you can define proper multiplication to build on that group to get very high in the algebraic structures tree. But in any case, this is deep within mathematics and I kinda doubt it would get you far in physics.

    Wo-how! Easy there. Set of all integrable function is much larger than set of differentiable functions and kinda sure that, if you try to build 1-to-1 relation between those, you'd run into world of pain. But go ahead and try :D

    You are kinda jiggling terms in the way that would cause a normal mathematician a epileptic attack. :) When you say smooth, you are already implying differentiable properties. If you are thinking continuous function, there are functions that are continuous everywhere, but differentiable nowhere.

    Well, with no offense meant - you actually didn't deliver any idea. Many of the terms in your opening post have very strict meaning and you can't just glue them by axiom. Actually, you can, but if you have a false axiom - you can prove anything. :D
     
  7. Jun 10, 2012 #6
    Have you been on a train, plane car, boat, ammusement park ride lately.
    Higher derivatives are taken into account in design of roads, railroad tracks, ..
    The next higher term is called jerk, and the next higher is jounce.

    A railway track does not go from a straight section to an immediate section of a circle. Doing so would cause an abupt acceleration as the train goes around the bend. The curve in the track is instead designed so the acceleration increases slowly from 0 at the straight section to a certain value at the midsection of the curve. Doing so allows the cargo to not tumble off the cars or the passengers to experience an umpleasant ride.

    Ammusemant park rides provide for some pretty wild and crazy movements. Higher derivatives are taken into account so the riders do not come off suffering from whiplash.

    In some systems the higher derivatives run out and become 0.
    In others such as a harmonic system with sin terms the derivatives are endless and you can end up with acceleration, jerk, jounce, snap, crackle, pop, ... although there is no standard as to what the higher terms are called. Jerk can also be referred to as jolt.
     
  8. Jun 10, 2012 #7
    It may interest you to know that in something like what you are asking already exists in the theory of bending


    [tex]\begin{array}{l}
    Displacement = y(x) \\
    Slope = \frac{{dy}}{{dx}} \\
    Moment = \frac{{{d^2}y}}{{d{x^2}}} \\
    Shear = \frac{{{d^3}y}}{{d{x^3}}} \\
    Load = \frac{{{d^4}y}}{{d{x^4}}} \\
    \end{array}[/tex]

    With the provision of some appropriate constants.

    So the load is the fourth order derivative of the displacement, or in order to obtain the diplacement from the load you have to integrate four times or solve a fourth oder diferential equation.
     
  9. Jun 10, 2012 #8

    I really didn't know that something called Frechet-space exists. I am really thankful for your information
     
  10. Jul 3, 2012 #9
    When a body free falls towards the Earth it possesses some acceleration 'g'( gravitational acceleration) ! but it decreases as the body falls toward the Earth because g=GM/R (as the body falls towards the Earth R decreases and g increases) ! ie. there is a rate of change of acceleration and its value is nagative as the acceleration goes on decreasing ! The concept of higher derivatives also occurs in mechanics and electronics in our three dimensional space ! displacement-velocity-jerk-jounce(snap)-crackle-pop-...........................
    Momentum equals mass times velocity!
    Force equals mass times acceleration!
    Yank equals mass times jerk!
    Tug equals mass times (jounce)snap!
    Snatch equals mass times crackle!
    Shake equals mass times pop!!
     
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