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

TL;DR

This paper develops a Hamiltonian framework for a symmetric top in axially symmetric external fields, analyzing stability and describing a broad class of interaction models, with specific application to the levitating Orbitron.

Contribution

It introduces a Hamiltonian model for symmetric tops in external fields with axial symmetry, including stability analysis of the levitating Orbitron.

Findings

Stability of the levitating Orbitron in relative equilibrium proved

Hamiltonian structure derived via reduction from $T^*SE(3)$

Model encompasses a wide class of interaction scenarios

Abstract

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.