Hamiltonian dynamics of a symmetric top in external fields having axial symmetry. Levitating Orbitron

The symmetric top is a special case of the general top, and canonical Poisson structure on $T^*SE(3)$ is the common method of its description. This structure is invariant under the right action of $SO(3)$, but the Hamiltonian of the symmetric top is invariant only under the right action of subgroup $S^1$ that corresponds to the rotation around the symmetry axis of the symmetric top. So, its Poisson structure was obtained as the reduction $T^*SE(3)/S^1$. Next we propose the Hamiltonian that describes the wide class of the interaction models of symmetric top and axially-symmetric external field. The stability of the levitating Orbitron in relative equilibrium was proved.