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A The Lagrange Top

  1. Mar 27, 2017 #1
    All the needed formulas are here http://hepweb.ucsd.edu/ph110b/110b_notes/node36.html
    I consider the following case
    $$p_\psi\ne 0,\quad p_\phi/p_\psi\in (\cos\theta_2,\cos\theta_1)$$ this case corresponds to the middle picture in the bottom of the cited page.

    I can not prove that the time average of the angle ##\phi## is not equal to zero: ##\int_0^\tau\phi(t)dt\ne 0##, here ##\tau## is the period of the function ##\theta(t)##.
    I know it looks like a standard simple thing but I have been thinking for three days and the result is zero, I also can not find it in books. Please help.
     
    Last edited: Mar 27, 2017
  2. jcsd
  3. Mar 27, 2017 #2

    Charles Link

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    Very interesting problem. I think I have a qualitative type proof, but it doesn't use the Lagrange equations. The top has a torque on it from gravity. This torque when integrated over time from ## 0 ## to ## \tau ## will be non-zero. (The ## \vec{r} ## and the ## \vec{F} ## don't do a cycle around the top during the time interval of one loop=they basically remain in the same general vicinity for the single loop.) This non-zero result means that the integral of ## \frac{d \vec{L}}{dt} ## (## \vec{L} ## is the angular momentum, basically from the spinning top) must be non-zero so that ## \Delta \vec{L} ## is non-zero. The top can not return to the same location after doing a small loop, because this would imply ## \Delta \vec{L}=0 ##.
     
    Last edited: Mar 27, 2017
  4. Mar 28, 2017 #3
    yes but the vector ##\boldsymbol L## is not parallel to the axis ##z_B##

    UPD: the vectors ##\boldsymbol L(t)## and ##\boldsymbol L(t+\tau)## can have the same direction but different value
     
    Last edited: Mar 28, 2017
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