Stability of Hamiltonian relative equilibria in symmetric magnetically confined rigid bodies

Abstract

This work studies the symmetries, the associated momentum map, and relative equilibria of a mechanical system consisting of a small axisymmetric magnetic body-dipole in an also axisymmetric external magnetic field that additionally exhibits a mirror symmetry; we call this system the ``orbitron. We study the nonlinear stability of a branch of equatorial relative equilibria using the energy-momentum method and we provide sufficient conditions for their $\mathbb{T}^2$--stability that complete partial stability relations already existing in the literature. These stability prescriptions are explicitly written down in terms of some of the field parameters, which can be used in the design of stable solutions. We propose new linear methods to determine instability regions in the context of relative equilibria that allow us to conclude the sharpness of some of the nonlinear stability conditions obtained.