# Euler's equation for rigid body rotation applied to inertia frame, with body frame both rotated and translated with respect to the world frame [duplicate]

The Euler equation is usually expressed in a local body frame: $$τ_b=I_b \dot{\vec \omega}_b + \vec \omega_b \times I_b \vec \omega_b$$ Where the subscript $$b$$ indicates that the respective term is expressed in the body frame $$b$$. Assuming the body frame is both rotating and translating (over time) with respect to the inertia frame, how can we express the Euler equations in the inertial (fixed) frame?

Related to the previous question, in such a case, how would the Inertia tensor expressed in the inertial frame ($$I_I$$) be related to the constant one expressed in the body frame ($$I_b$$)?

This thread discusses a related question, however, it seems that the answer only treats the case where the body frame is rotating with respect to the inertial frame.

If possible please include references to detailed discussions on the topic and how the result is derived.

• You can also give this discussion of mine a try: physics.stackexchange.com/a/568099/118976 Mar 14, 2021 at 2:34
• @JohnAlexiou Yes it does, thank you!
– Elij
Mar 14, 2021 at 14:36
• @Futurologist that's helpful as well thanks!
– Elij
Mar 14, 2021 at 14:37