# Motion of a rigid body with constant angular momentum

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?

• 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 9 '16 at 0:29
• You mean rotational velocity vector right? – ja72 Dec 9 '16 at 12:36
• 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. – ja72 Dec 9 '16 at 12:44
• 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 9 '16 at 14:16
• ja72: yes, good point! I edited the question to make that clear. – Martin C. Martin Dec 9 '16 at 14:17