On the stability of the permanent rotations of a charged rigid body-gyrostat

Abstract

We consider the motion of a charged rigid body about a fixed point carrying a rotor that is attached along one of the principal axes of the body. This motion occurs under the action of the resultant of the uniform gravity field and the homogeneous magnetic field. The equations of motion are formulated, and they are presented by means of the Hamiltonian function in the framework of the Lie–Poisson system. These equations of motion have six equilibrium solutions. The sufficient conditions for instability for these equilibria are studied by utilizing the linear approximation method, while the sufficient conditions for stability are presented by means of the energy-Casimir method. For certain configuration of the body, the regions of Lyapunov stability and instability are determined in the plane of some parameters. Furthermore, we clarify that the regions of Lyapunov stability are a portion of the regions of linear stability.