Euler angles in torque free precession of a symmetric top

[riveay]

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!

 

[vanhees71]

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

 

[riveay]

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

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.

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

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.

 

[vanhees71]

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.