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Concept# Euler's equations (rigid body dynamics)

Summary

In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with angular velocity ω whose axes are fixed to the body. Their general vector form is
:
\mathbf{I} \dot{\boldsymbol\omega} + \boldsymbol\omega \times \left( \mathbf{I} \boldsymbol\omega \right) = \mathbf{M}.
where M is the applied torques and I is the inertia matrix.
The vector \dot{\boldsymbol\omega} is the angular acceleration. Again, note that all quantities are defined in the rotating reference frame.
In orthogonal principal axes of inertia coordinates the equations become
:
\begin{align}
I_1,\dot{\omega}*{1} + (I_3-I_2),\omega_2,\omega_3 &= M*{1}\
I_2,\dot{\omega}*{2} + (I_1-I_3),\omega_3,\omega_1 &= M*{2}\
I_3,\dot{\omega}*{3} + (I_2-I_1),\omega_1,\omega_2 &= M*{3}
\end{align}
where Mk are the components of the applied torques, Ik

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