Mechanics of Continua and Structures
In this section we will be deriving the euler equations. These equations are a consequence of the balance of angular momentum. The balance of angular momentum follows from Newton’s second law of motion. One form of the statement of balance of angular momentum for a rigid solid reads,
\[\begin{equation} \dot{\bs{H}}_t=\bs{M}, \end{equation}\]where $\bs{H}_t$ is the angular momentum of the solid, and the $\bs{M}$ is the resultant external moment on the solid w.r.t its center of mass. That is, if a set of forces $ \bs{f}_i$ acts on the solid at the points whose position vectors are $\bs{x}_i$, then
\(\begin{align} \bs{M}&=\bs{\pi}_i \times \bs{f}_i, \end{align}\) where $\bs{\pi}_i=\bs{x}_i-\bar{x}$
In a previous section we showed that
\[\begin{align} \dot{\bs{H}}_t=&\left(\lambda_1 \dot{\omega}_1-(\lambda_2-\lambda_3)\omega_2 \omega_3\right)\uv{e}_1+\\ &\left(\lambda_2 \dot{\omega}_2-(\lambda_3-\lambda_1)\omega_1 \omega_3\right)\uv{e}_2+\\ &\left(\lambda_3 \dot{\omega}_3-(\lambda_1-\lambda_2)\omega_1 \omega_2\right)\uv{e}_3\\ \end{align}\]The balance of angular momentum thus implies that
\[\begin{align} \lambda_1 \dot{\omega}_1-(\lambda_2-\lambda_3)\omega_2 \omega_3&=\bs{M}\cdot \uv{e}_1,\\ \lambda_2 \dot{\omega}_2-(\lambda_3-\lambda_1)\omega_1 \omega_3&=\bs{M}\cdot \uv{e}_2,\\ \lambda_3 \dot{\omega}_3-(\lambda_1-\lambda_2)\omega_1 \omega_2&=\bs{M}\cdot \uv{e}_3,\\ \end{align}\]