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When the angular velocity vector isn't in the same direction as the angular momentum (measured about the center of mass), the angular velocity will change direction over time. Is there a closed form for angular velocity and/or orientation over time?

Martin C. Martin
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    It is not clear to me what you are asking about. If angular momentum $L$ is constant and the body is rigid (ie moment of inertia $J$ is constant) then the angular velocity $\omega$ is also constant since $L=J\omega$. – sammy gerbil Dec 09 '16 at 00:29
  • You mean rotational velocity vector right? – John Alexiou Dec 09 '16 at 12:36
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    I think you are asking for closed form solutions of $${\rm I} \dot{\vec{\omega}} + \vec{\omega} \times {\rm I} \vec{\omega} =0$$

    I don't know of any, but someone may have some better insight into this.

     – John Alexiou Dec 09 '16 at 12:44
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    sammy gerbil: The moment of inertia is constant in the body frame, but changes direction in the inertial frame. A cylinder whose long direction is parallel to the $x$ axis, compared to parallel to the $y$ axis, will have the first two columns of the matrix swapped. So as the body rotates, the moment of inertia rotates, meaning the angular velocity must also change in order to keep the angular momentum $L$ constant. – Martin C. Martin Dec 09 '16 at 14:16
  • ja72: yes, good point! I edited the question to make that clear. – Martin C. Martin Dec 09 '16 at 14:17
  • @ja72 Yep I think that's the equation, thanks for pointing it out. Maybe I'll ask on the math stack exchange now that I have that. Cheers! – Martin C. Martin Dec 09 '16 at 14:28
  • Yes, and remember that with a body rotation matrix ${\rm R}$ the MMOI matrix is $${\rm I} = {\rm R} ,{\rm I}_{\rm body},{\rm R}^\top$$ – John Alexiou Dec 09 '16 at 16:23
  • Let us continue this discussion in chat. – Martin C. Martin Dec 09 '16 at 17:27
  • Related post: http://physics.stackexchange.com/a/113896/392 – John Alexiou Dec 09 '16 at 22:13

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I think you might be interested in the Poinsot constructions, see the corresponding Wikipedia article for an overview. For a slightly different angle at an answer to this question, have a look at this Mathematica Demonstration. In a similar vein, there's a fairly nice simulation here.

Long story short, there's no convenient closed-form solutions for the general case, but numerical solutions can be generated readily.

Addendum: The motion can, in principle, be described using Jacobian elliptic functions. This was published by Jacobi in 1850, in his paper "Sur la rotation d’un corps", in the German Journal für die reine und angewandte Mathematik 39:293–350. The PDF of the original paper is available here.

More references: A related question was asked on this forum, here. This document has some more suggestions for algorithms to describe rigid-body rotations. Googling "rigid body dynamics Jacobian elliptic functions" will reveal a wealth of additional information.

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