I was asked to formulate the equations governing the rotation of a body moving without any external moments acting about its centre of mass in terms of a coupled system of first order, nonlinear differential equations. I decided to go with the Euler equations, and I ended up with this:
\begin{equation} \label{symdif}
\begin{array}{l l l}
\dot{\omega}_x=\frac{I_{yy}-I_{zz}}{I_{xx}}\omega _y\omega _z\\
\dot{\omega}_y=\frac{I_{zz}-I_{xx}}{I_{yy}}\omega _z\omega _x\\
\dot{\omega}_z=\frac{I_{xx}-I_{yy}}{I_{zz}}\omega _x\omega _y
\end{array}
\end{equation}
\begin{equation}
(I_{xx}=I_{yy}<I_{zz})
\end{equation}
This indicates that $$\omega_z=constant$$which makes it possible to solve the system of differential equations, but I wonder how one would end up with the differential equations explicitly asked for.
\begin{equation} \label{symdif}
\begin{array}{l l l}
\dot{\omega}_x=\frac{I_{yy}-I_{zz}}{I_{xx}}\omega _y\omega _z\\
\dot{\omega}_y=\frac{I_{zz}-I_{xx}}{I_{yy}}\omega _z\omega _x\\
\dot{\omega}_z=\frac{I_{xx}-I_{yy}}{I_{zz}}\omega _x\omega _y
\end{array}
\end{equation}
\begin{equation}
(I_{xx}=I_{yy}<I_{zz})
\end{equation}
This indicates that $$\omega_z=constant$$which makes it possible to solve the system of differential equations, but I wonder how one would end up with the differential equations explicitly asked for.