Integrability, localisation and bifurcation of an elastic conducting rod in a uniform magnetic field

Motivated by the problem of electrodynamic space tethers, the equilibrium equations for an elastic conducting rod in a magnetic field are investigated. In body coordinates the equations are found to sit in a family of noncanonical Hamiltonian systems. These systems, which include the classical Euler and Kirchhoff rods, are shown to be completely integrable under certain material conditions. Mel’nikov method is then used to show that two integrable perturbations can in fact destroy the integrable structure, resulting in multimodal rod configurations. Such solutions are investigated numerically revealing a rich bifurcation structure.