# Euler's Equations of Motion for Rotation of Rigid Body

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## Theorem

Let a rigid body $B$ rotate about an axis $A$ which is fixed in relation to $B$ and parallel to the principal axis of inertia of $B$.

Then the rotation of $B$ about $A$ is described by:

- $\mathbf I \cdot \dot {\boldsymbol \omega} + \boldsymbol \omega \times \left({\mathbf I \cdot \boldsymbol\omega}\right) = \mathbf M$

where:

- $\mathbf M$ is the applied torque applied to $B$ about $A$
- $\mathbf I$ is the moment of inertia of $B$ with respect to $A$
- $\boldsymbol \omega$ is the angular velocity about $A$.

## Proof

## Source of Name

This entry was named for Leonhard Paul Euler.

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3$: Appendix $\text A$: Euler - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.21$: Euler ($\text {1707}$ – $\text {1783}$)