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Helical buckling of a whirling conducting rod in a uniform magnetic field

We study the effect of a magnetic field on the behaviour of a conducting elastic rod subject to a novel set of boundary conditions that, in the case of a transversely isotropic rod, give rise to exact helical post-buckling solutions. The equations used are the geometrically exact Kirchhoff equations and both static (buckling) and dynamic (whirling) instability are considered. Critical loads are obtained explicitly and are given by a surprisingly simple formula. By solving the linearised equations about the (quasi-)stationary solutions we also find secondary instabilities described by (Hamiltonian-)Hopf bifurcations, the usual signature of incipient `breathing' modes. The boundary conditions can also be used to generate and study helical solutions through traditional non-magnetic buckling due to compression, twist or whirl.