91Ó°ÊÓ

We start by writing the Euler's equations of motion for a rigid body in the principal axes frame. \( \bold{I}_1 \dot{\omega_1} - \omega_2 \omega_3 (\bold{I}_2 - \bold{I}_3) = 0 \), \( \bold{I}_2 \dot{\omega_2} - \omega_3 \omega_1 (\bold{I}_3 - \bold{I}_1) = 0 \), \( \bold{I}_3 \dot{\omega_3} - \omega_1 \omega_2 (\bold{I}_1 - \bold{I}_2) = 0 \), Here, \({ \bold{I} }_1\), \({ \bold{I} }_2\), and \({ \bold{I} }_3\) are the principal moments of inertia, and \({ \omega_1 }\),\({ \omega_2 }\), \({ \omega_3 }\) are the components of the angular velocity vector in the principal axes frame.

Let's first consider the rotation about the largest principal axis. This corresponds to \({ \omega_1 } = 0\), \({ \omega_2 } = 0\), and \({ \omega_3 } \neq 0\). We can rewrite the Euler's equations as: \( \bold{I}_1 \dot{\omega_1} = \omega_2 \omega_3 (\bold{I}_2 - \bold{I}_3) \), \( \bold{I}_2 \dot{\omega_2} = \omega_3 \omega_1 (\bold{I}_3 - \bold{I}_1) \), \( \bold{I}_3 \dot{\omega_3} = \omega_1 \omega_2 (\bold{I}_1 - \bold{I}_2) \). We see that: \( \dot{\omega_1} = 0 \), \( \dot{\omega_2} = 0 \), \( \dot{\omega_3} = 0 \), The system is at rest, as the derivatives of \(\omega_1\) and \(\omega_2\) are equal to zero. Now let's consider rotation about the smallest principal axis. In this case, \({ \omega_1 } \neq 0\), \({ \omega_2 } = 0\), and \({ \omega_3 } = 0\). We can rewrite the Euler's equations as: \( \bold{I}_1 \dot{\omega_1} = \omega_2 \omega_3 (\bold{I}_2 - \bold{I}_3) \), \( \bold{I}_2 \dot{\omega_2} = \omega_3 \omega_1 (\bold{I}_3 - \bold{I}_1) \), \( \bold{I}_3 \dot{\omega_3} = \omega_1 \omega_2 (\bold{I}_1 - \bold{I}_2) \). We see that: \( \dot{\omega_1} = 0 \), \( \dot{\omega_2} = 0 \), \( \dot{\omega_3} = 0 \). Again, this system is at rest, as the derivatives of \(\omega_2\) and \(\omega_3\) are equal to zero.

For a system expressed by the Euler's equations, Liapunov stable equilibrium implies that the equilibrium is stable under small perturbations. We will prove Liapunov stability if the system returns to rest after being slightly perturbed. For rotations about the largest principal axis, any small perturbation will have nonzero components of \(\omega_1\) and \(\omega_2\). However, their derivatives are zero (as we found in Step 2). Therefore, the system will return to rest, and the rotation about the largest principal axis is Liapunov stable. For rotations about the smallest principal axis, any small perturbation will have nonzero components of \(\omega_2\) and \(\omega_3\). However, their derivatives are zero (as we found in Step 2). Therefore, the system will return to rest, and the rotation about the smallest principal axis is also Liapunov stable. In conclusion, stationary rotations of a rigid body around both the largest and the smallest principal axes are Liapunov stable.