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I Euler angles in torque free precession of a symmetric top

  1. Dec 1, 2017 #1
    Is calculating the Euler angles analitically possible?

    I am trying to obtain the angles to transform the body-fixed reference frame to the inertial reference frame. I can get them without problems with numerical methods. But I would to validate them analitically, if possible.

    I followed the steps by Landau & Lifshitz (https://archive.org/stream/Mechanics_541/LandauLifshitz-Mechanics#page/n123/mode/2up) and found the angular velocity in the body frame. Which is also here.

    Now, I understand that when the angular momentum vector is aligned with the inertial Z axis, then the angle rates are:

    $$ \dot{\theta} = 0 $$ $$ \dot{\phi} = M/I_1 $$ $$ \dot{\psi} = M\cos \theta (1/I_3 - 1/I_1) $$

    But what if the angular momentum and the Z axes are not aligned? When this happens, ##\theta## stops being constant, doesn't it?

    Thank you in advance!
     
  2. jcsd
  3. Dec 2, 2017 #2

    vanhees71

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    The angular momentum of the spinning top is constant in the inertial frame, and for the standard Euler angles, as depicted in

    http://theory.gsi.de/~vanhees/faq/mech/node22.html

    it's most convenient to choice the basis fixed in the inertial system such that the angular momentum is pointing in ##z## direction.

    You can find a complete treatment in a mixed form using both the Euler equations for the free top (non-holonomic coordinates) and the Euler angles (holonomic coordinates) in (sorry, I have this written up in German only yet):

    http://theory.gsi.de/~vanhees/faq/mech/node78.html
     
  4. Dec 3, 2017 #3
    I understand. My problem is that I'm trying to validate the results of a simulation that is constrained to the Euler angle equations where both the Z and e3 axes are parallel.

    My german is not that good, but from what I understand, your approach also aligns the Z axis with the constant angular momentum vector and derives the angular velocity from it, doesn't it?

    I find that approach interesting.

    Thanks for the help.
     
  5. Dec 4, 2017 #4

    vanhees71

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    Yes, sure. The reason is that the choice of the ##z##-axis of the inertial system in direction of the angular momentum (which is conserved in this system) is particularly convenient, because of the choice of the ##3## axis in the rotations defining the Euler angle. For the same reason, it's also convenient to put the figure axis of the symmetric top in the direction of the ##z'##-axis of the body-fixed frame.
     
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