Lagrange top

Example

The Lagrange top is a heavy rigid body turning about a fixed point $O$ in a uniform gravitational field. It is symmetric about an axis through $O$. In the principal body frame at $O$, its moments of inertia are $I_1 = I_2$ and $I_3$, and its centre of mass is at $\ell e_3$ with $\ell \ne 0$. Let $m$ be its mass, $g$ the gravitational acceleration, $\omega$ the angular velocity and $\gamma$ the upward unit vertical, both in the body frame. The motion is given by the Euler–Poisson equations $$I\dot\omega = (I\omega) \times \omega + m g \ell\, \gamma \times e_3, \qquad \dot\gamma = \gamma \times \omega,$$ where $I = \operatorname{diag}(I_1, I_1, I_3)$.

It is an integrable system on the $6$-dimensional phase space of the heavy top, with three integrals in involution:

1. the energy $H = \tfrac12 \omega^T I \omega + m g \ell\, \gamma_3$;
2. the vertical component $L \cdot \gamma$ of the angular momentum $L = I\omega$;
3. the spin $\omega_3$ about the symmetry axis.

The check for the third: the third component of $(I\omega) \times \omega$ is $I_1\omega_1\omega_2 - I_1\omega_2\omega_1 = 0$, and the third component of $\gamma \times e_3$ is $0$, so $I_3\dot\omega_3 = 0$.