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According to wikipedia page https://en.wikipedia.org/wiki/Angular_velocity Euler proved that the projections of the angular velocity pseudovector on each of these three axes is the derivative of its associated angle (which is equivalent to decomposing the instantaneous rotation into three instantaneous Euler rotations). Therefore:

$\boldsymbol{\omega}=\dot{\alpha}\boldsymbol{u}_1+\dot{\beta}\boldsymbol{u}_2+\dot{\gamma}\boldsymbol{u}_3$

Where we can get:

$\dot{\alpha}=\frac{\boldsymbol{u}_2 \times \boldsymbol{u}_3}{(\boldsymbol{u}_2 \times \boldsymbol{u}_3)\cdot\boldsymbol{u}_1}\boldsymbol{\omega}$

$\dot{\beta}=\frac{\boldsymbol{u}_1 \times \boldsymbol{u}_3}{(\boldsymbol{u}_1 \times \boldsymbol{u}_3)\cdot\boldsymbol{u}_2}\boldsymbol{\omega}$

$\dot{\gamma}=\frac{\boldsymbol{u}_1 \times \boldsymbol{u}_2}{(\boldsymbol{u}_1 \times \boldsymbol{u}_2)\cdot\boldsymbol{u}_3}\boldsymbol{\omega}$

Is it possible to get angles $\alpha$,$\beta$,$\gamma$, knowing only $\boldsymbol{\omega}$,$\boldsymbol{u}_1$,$\boldsymbol{u}_2$,$\boldsymbol{u}_3$?

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