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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


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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


    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):

  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


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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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