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| We study the symmetries, the associated momentum map, and relative equilibria of mechanical systems consisting of magnetically confined rigid bodies in axisymmetric external magnetic fields; some instances of such systems include the so called orbitrons, levitrons, and others. We study the nonlinear stability of branches of relative equilibria using the energy-momentum method and we provide sufficient conditions for the existence of $G_{\mu}$-stability. These stability prescriptions are explicitly written down in terms 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 we use to conclude the sharpness of some of the nonlinear stability conditions obtained. The results that we present are proved to be generalizable to other specific symmetric systems of magnetically confined rigid bodies. |
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