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

## 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 $A$.

## Source of Name

This entry was named for Leonhard Paul Euler.