Integrability of a conducting elastic rod in a magnetic field

Abstract

Abstract. We consider the equilibrium equations for a conducting elastic rod placed in a uniform magnetic field, motivated by the problem of electrodynamic space tethers. When expressed in body coordinates the equations are found to sit in a hierarchy of noncanonical Hamiltonian systems involving an increasing number of vector fields. These systems, which include the classical Euler and Kirchhoff rods, are shown to be completely integrable in the case of a transversely isotropic rod; they are in fact generated by a Lax pair. For the magnetic rod this gives a physical interpretation to a previously proposed abstract nine-dimensional integrable system. We use the conserved quantities to reduce the equations to a four-dimensional canonical Hamiltonian system, allowing the geometry of the phase space to be investigated through Poincaré sections. In the special case where the force in the rod is aligned with the magnetic field the system turns out to be superintegrable, meaning that the phase space breaks down completely into periodic orbits, corresponding to straight twisted rods.

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

“…Steadily rotating (whirling) solutions satisfy the equations (14) and (15) with the dotted variables set to zero:…”

Spatial chaos of an extensible conducting rod in a uniform magnetic field

“…The existence of transverse homoclinic orbits implies that the 'dynamics' near the hyperbolic saddle is 'chaotic' in the sense that the following holds: . The reduction follows [6] but now allows for extensibility and shearability of the rod. The reduction in [6] was shown to be canonical on the condition that the force n and magnetic fieldB are not aligned.…”

“…For the force we can write n = R (q) w (q, p), for some non-constant triple w. By decomposing w into parts perpendicular and parallel to k and using the Casimirs (2.18) and (2.19) we obtain [6]…”

“…It is also known that some perturbations of the rod equations are integrable, but that others are not. For instance, anisotropy of the cross-section [2] and intrinsic curvature [3] destroy integrability, as does the effect of gravity [4], but extensibility and shearability [5] do not, nor does the effect of an external force due to a uniform magnetic field [6].…”

Nonintegrability of an extensible conducting rod in a uniform magnetic field

“…It is also known that some perturbations of the rod equations are integrable, but that others are not. For instance, anisotropy of the cross-section [15] and intrinsic curvature [2] destroy integrability, as does the effect of gravity [6], but extensibility and shearability [18] do not, nor does the effect of an external force due to a uniform magnetic field [16].…”