#physics
An attempt to model the full, free-body equations for motion of a rigid body.

We know $$\vec{\tau} = \left( \frac{d\mathbf{L}}{dt} \right)_{I}$$ in that inertial frame.

Applying the concept of Transport Theorem, we can write $$\vec{\tau} = \left( \frac{d\mathbf{L}}{dt} \right)_{B} + \vec{\omega} \times \mathbf{L}$$
where $\overrightarrow{\omega }$ is the angular velocity of the body in the inertial frame.

We consider the principal axes to model our body around, so our inertia matrix is simplified to $$\pu{diag}[I_{xx}\ \ \ I_{yy}\ \ \ I_{zz}]$$
Thus the components of $\mathbf{L}$ are $$\mathbf{L} = I_{xx}\omega_{x} + I_{yy}\omega_{y} + I_{zz}\omega_{z}$$

Therefore, evaluating the expression given to us by the Transport theorem, we note that $\mathbf{I}$ does not change in the body frame, and therefore

$$\begin{align} \tau_{x} &= I_{xx}\dot{\omega}{x} - (I - I_{zz})\omega_{y}\omega_{z} \ \tau_{y} &= I_{yy}\dot{\omega}{y} - (I - I_{xx})\omega_{z}\omega_{x} \ \tau_{z} &= I_{zz}\dot{\omega}{z} - (I - I_{yy})\omega_{x}\omega_{y} \end{align}$$

These 3 equations are together called "Euler's equations of motion"

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