Heavy symmetric top Lagrangian

For a fixed point a distance $a$ from the centre of mass and inclination measured from the upward vertical,The spin and precession angles are cyclic.

A Lagrange top is an axisymmetric rigid body with one point fixed on its symmetry axis, moving in a uniform gravitational field with its center of mass on that axis.

For a Lagrange top whose inclination $\theta$ is measured from the upward vertical, steady precession at angular speed $\Omega$ obeysThus steady precession at a prescribed $0<\theta <\pi /2$ exists precisely when

With transverse and axial moments $I,J$, the Hamiltonian is

Fixing the cyclic momenta $p_{\psi }=\lambda$ and $p_{\varphi }=\mu$ leaves the canonical pair $(\theta ,p_{\theta })$ and a one-degree-of-freedom Hamiltonian with effective potential

At the upright spinning state, $p_{\varphi }=p_{\psi }=\lambda$. The reduced potential is locally minimized at zero inclination whenso sufficiently large axial angular momentum stabilizes the gravitationally inverted top.