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I Why do some but not all derivatives have physical meaning?

  1. Oct 19, 2016 #1
    I know that taking the derivative of certain functions that explain physical phenomena can lead to another statement describing the physical system, the most famous being the derivatives of position. That is,
    position-->velocity-->acceleration-->jerk-->jounce...and taking any other further derivatives suddenly becomes physically meaningless. Is there any intuitive way of thinking about the "limits" of derivatives when it comes to describing physical or geometric systems?
     
  2. jcsd
  3. Oct 19, 2016 #2
    What makes you say the derivative of jounce is physically meaningless? Is it physically meaningless just because you don't have a name for it?
     
  4. Oct 19, 2016 #3
    I guess what I meant by meaningless is when it fails to have applications in terms of describing a physical system.
    Do you know of any applications of the derivative of jounce?
     
  5. Oct 19, 2016 #4
    Lack of applications isn't the same as meaningless. And it's subjective.
     
  6. Oct 19, 2016 #5

    A.T.

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    Suddenly? Doesn't the range of applications decrease with each derivative?
     
  7. Oct 19, 2016 #6
    What would be the meaning if I integrated the volume of a sphere?

    V=4/3pir^3 -----> 4/9pi^r4

    What does this integral describe if anything?
     
  8. Oct 19, 2016 #7
    I messed up..the integral would be 1/3pir^4 +c
     
  9. Oct 19, 2016 #8
    You aren't asking meaningful questions.
     
  10. Oct 19, 2016 #9
    I'm sorry for the vagueness.
    I know that the derivative of the volume of the sphere is equal to the surface area of the sphere. That is intuitive to imagine, because with each infinitesimal change in the radius, it's like adding an infinitely thin coat of paint on the outside of the sphere, which would be the surface area. Also I note that if I integrate geometric shapes, one dimension is added.
    Now lets say I were to integrate the volume of a sphere, would this give me a description of a 4 dimensional object, and if so, is it only unintuitive to us because we live in a 3 dimensional world? Or can higher dimensions be utilized in mathematics but not physics?
     
  11. Oct 19, 2016 #10
    I have heard there are models describing n dimensions, usually would be associated with topology in the mathematics field, however. The integral you speak of in the quote would describe a 4-D volume, of infinitesimally summed together 3-D volumes. Any topologists that could give the proper term would be nice.

    Anyway, it is not that each mathematical operation makes it physically meaningless. The meaning is already put into the equations, and that is by defining x and t. For the derivatives above:
    x - position, t- Time
    dx/dt = velocity = v - change in position x over change in time.
    dv/dt = acceleration = a - change in the velocity.
    da/dt = jerk = j - change in acceleration
    and so on.

    The mathematics may blur out the physics, but once you have a model described as differential equations, with each parameter and variable and function defined, the physics gets carried to the end result and would produce a relationship between them all, that would have a physical meaning.
    Basically, if you have a meaning in the equation to begin with, the is no reason it would be lost.
     
  12. Oct 19, 2016 #11
    A very fundamental issue which is not well taught. Not only does mathematics stand alone, if we lived in a very different universe, the utility of particular parts of math may be more or less useful, but math is not just separate from physics, when something is proved true in math, that's it. (Unless there is an error discovered later in the proof.) In physics, or any science, observations rule. Any beautiful theory can be destroyed by an ugly fact. More important, you may have a theory in physics that is believed true for hundreds of years before a contradiction is found.

    Now for a disturbing question. Is quantum mechanics math or physics? What about string theory? Right now the best we can say is that there is a mathematics called QM which seems to correspond to the real world (actually QCD now). The same may be true of string theory, and even it may replace QM (technically as a physical theory) but the mathematics developed to support QM will still be true, even if string theory replaces QCD in physics. The disturbing question? Why does this math work so well?
     
  13. Oct 19, 2016 #12
    The cam shaft in your car is designed using higher derivatives of position. Jounce or jerk is used at the peak and foot of the cam where the acceleration changes sign, higher derivatives are involved as well.
     
  14. Oct 20, 2016 #13

    David Lewis

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    Quantum Mechanics might be something in between, like hyper-applied mathematics.
     
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