**The Euler equations**

**Euler equation inertial frame at the CM**
\begin{align*}
&\frac{d}{dt}\left(\mathbf I\,\mathbf\omega\right)=\mathbf\tau\\
&\Rightarrow\\
&\mathbf I\,\mathbf{\dot{\omega}}+\mathbf{\dot{I}}\,\mathbf\omega=\mathbf\tau\quad
,\text{with}\\
&\mathbf I=\mathbf R\,\mathbf I_B\quad ,\mathbf{\dot{I}}=\mathbf{\dot{R}}\,\mathbf I_B
= 
 \underbrace{\left[ \begin {array}{ccc} 0&-\omega_{{z}}&\omega_{{y}}
\\  \omega_{{z}}&0&-\omega_{{x}}\\
-\omega_{{y}}&\omega_{{x}}&0\end {array} \right]}_{\mathbf{\omega}^\times}
\,\mathbf R\,\mathbf I_B\quad\Rightarrow\\\\
&\boxed{\quad\mathbf I\,\mathbf{\dot{\omega}}+\mathbf\omega\times\,\mathbf I\,\mathbf\omega=\mathbf\tau\quad~Eq. (1)}
\end{align*}

**Euler equation body  frame at the CM**

\begin{align*}
 &\text{with}\\
 &\mathbf I=\mathbf{R}\,\mathbf I_B\,\mathbf R^T\quad,\text{you obtain Eq. (1)}\\
 &\mathbf{R}\,\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf{\dot{\omega}}}_{ \mathbf{\dot{\omega}}_B}
 +\mathbf\omega\times\, \mathbf{R}\,\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf\omega}_{ \mathbf\omega_B}=\mathbf\tau\quad,\text{ multiply from the left with}~\mathbf R^T\quad,\Rightarrow\\
 &\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf{\dot{\omega}}}_{ \mathbf{\dot{\omega}}_B}
 +\mathbf R^T\,\mathbf\omega\times\, \mathbf{R}\,\mathbf I_B\,\underbrace{\mathbf R^T\,\mathbf\omega}_{ \mathbf\omega_B}=\mathbf R^T\,\mathbf\tau\\\\
 &\boxed{\quad\mathbf I_B\,\mathbf{\dot{\omega}}_B+\mathbf\omega_B\times\,\mathbf I_B\,\mathbf\omega_B=\mathbf\tau_B\quad~Eq. (2)}
\end{align*}

Euler equation body  frame at body  point $~P$
\begin{align*}
 &\text{with }\\
 &\mathbf I_B=\mathbf I_P+\underbrace{m\, \left[ \begin {array}{ccc} 0&-z_{{{\it CP}}}&y_{{{\it CP}}}
\\  z_{{{\it CP}}}&0&-x_{{{\it CP}}}
\\  -y_{{{\it CP}}}&x_{{{\it CP}}}&0\end {array}
 \right]
\, \left[ \begin {array}{ccc} 0&-z_{{{\it CP}}}&y_{{{\it CP}}}
\\  z_{{{\it CP}}}&0&-x_{{{\it CP}}}
\\  -y_{{{\it CP}}}&x_{{{\it CP}}}&0\end {array}
 \right]}_{\mathbf I_{CP}}\\
 &\mathbf\omega_P=\mathbf\omega_B\\
 &\mathbf\tau_P=\mathbf\tau_B+\mathbf r_{CP}\times\mathbf F_F\quad\Rightarrow\\\\
  &\boxed{\quad\left(\mathbf I_P+\mathbf I_{CP}\right)\,\mathbf{\dot{\omega}}_P+\mathbf\omega_P\times\, \left(\mathbf I_P+\mathbf I_{CP}\right)\,\mathbf\omega_P=\mathbf\tau_P\quad~Eq. (3)}
\end{align*}

 - $~\mathbf I~$ Intertia tensor
 - $~\mathbf\tau~$ External torgue
 - $~\mathbf R~$ transformation matrix between body frame and inertial frame
 - $~\mathbf r_{CP}~$ body fixed vector from the center of mass to point p
 - $~\mathbf F_F~$ fictitious forces at the center of mass
 - $~$ subscript B body frame at the center of mass
 - $~$ subscript P body frame at body point P