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How to solve: ((1/k)')^2 + T^2(1/k)^2 - T^2 = 0

  1. Mar 11, 2009 #1
    That equation again is: ((1/k)')^2 + T^2(1/k)^2 - T^2 = 0
    where k is a function of s, T is a constant, and ' denotes differentiation with respect to s.

    If you're wondering, it arises when you try to describe the family of curves with constant torsion that lie on the unit sphere. Here, T is the torsion, and k is the curvature. And boy do I wish I remembered my differential equations class a little better.


    My attempt:

    Let 1/k = R.

    Then (R')^2 + T^2*R^2 -T^2 = 0

    Suppose R = A*cos(Ts) + B*sin(Ts)

    Plugging it in, it is found that A^2+B^2=1

    So, R = cos(a)*sin(Ts)+sin(a)+cos(Ts)

    So K = R^-1...

    Which does not satisfy the equation :(


    I also tried setting it up like this:

    K' +Sqrt[T^2*K^2*(K^4-K^2)]
     
  2. jcsd
  3. Mar 11, 2009 #2

    tiny-tim

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    Hi Pwantar! :smile:
    Nooo …

    what are the roots of the characteristic equation? :wink:
     
  4. Mar 11, 2009 #3

    HallsofIvy

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    That's a non-linear equation. Neither Acos(Ts)+ Bsin(Ts), nor the concept of "characteristic equation" apply.
     
  5. Mar 11, 2009 #4

    tiny-tim

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    oops!

    oops! thanks, HallsofIvy! :redface:

    (all those ^2s confusing me)

    better write it R'/√(1 - R2) = ±T :smile:
     
  6. Mar 11, 2009 #5
    As I'm too lazy to recheck what I've written now that I'm already at this fancy text box thing that I don't understand, but love, I would like to point out an error that may or may not be present, as well as a possible solution.

    ((1/k)')^2 + T^2*(1/k)^2 - T^2 = 0

    (-k'/k^2)^2 + T^2*(1/k)^2 - T^2 = 0

    (k')^2/k^4 + T^2*(1/k)^2 - T^2 = 0

    (k')^2 = T^2*k^2*(k^2-1)

    dk/ds = T*k*Sqrt(k^2-1)

    -atan(1/sqrt(k^2-1) = Ts + C, where C is an arbitrary constant

    k = sqrt(1/tan(-Ts-C)^2 + 1)

    k = 1/cos(Ts + C)

    which happily satisfies my equation... the thing I don't like is taking the square root. Perhaps I should consider both the negative and positive sqrts... however, just looking at it, I don't think the solution will change.


    One more thing....
    That Solution to the equation for R that I mentioned earlier actually does work... I guessed it based off of me forseeing a few sin^2 + cos^2 dealies cancelling out.

    the knowledge that k=1/R does not appear to be sufficient information to find k given R... however, should a = Pi/2, it works, save for the phase angle C. It could be that I made a mistake somewhere. Perhaps I'll actually try to find my work for it, examine it, and notify you of anything interesting.


    Whoops, I made some mistakes in what I typed, but got the answer... I'll leave it as an excercise to you to figure out what they are
     
    Last edited: Mar 11, 2009
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