Euler top

Example

The Euler top is a rigid body that turns about a fixed point $O$ with no external torque. This happens when $O$ is the centre of mass and gravity is the only force, or when no force acts at all. Let $I_1, I_2, I_3 > 0$ be its principal moments of inertia at $O$ and $\omega$ its angular velocity in the principal body frame. The motion of $\omega$ is given by Euler's equations with zero torque: $$I_1\dot\omega_1 = (I_2 - I_3)\omega_2\omega_3, \quad I_2\dot\omega_2 = (I_3 - I_1)\omega_3\omega_1, \quad I_3\dot\omega_3 = (I_1 - I_2)\omega_1\omega_2.$$

It is an integrable system. Its phase space is the cotangent bundle of the rotation group, of dimension $6$, so three independent integrals in involution are needed. These are the energy $H = \tfrac12(I_1\omega_1^2 + I_2\omega_2^2 + I_3\omega_3^2)$, the squared length $\abs{L}^2 = I_1^2\omega_1^2 + I_2^2\omega_2^2 + I_3^2\omega_3^2$ of the angular momentum, and one component of the angular momentum in a fixed frame. The check that $H$ is conserved: $\dot H = \sum_i I_i \omega_i \dot\omega_i = \omega_1\omega_2\omega_3\big((I_2 - I_3) + (I_3 - I_1) + (I_1 - I_2)\big) = 0$. The same computation with weights $I_i^2$ shows that $\abs{L}^2$ is conserved, and the fixed-frame angular momentum is conserved because no torque acts.