Integrability of a conducting elastic rod in a magnetic field

TL;DR

This paper investigates the integrability of a conducting elastic rod in a magnetic field, revealing complete integrability for isotropic rods and superintegrability in specific force alignments, with implications for space tether dynamics.

Contribution

It demonstrates the complete integrability of the magnetic rod equations using a Lax pair and reduces the system to a canonical Hamiltonian form for phase space analysis.

Findings

The system is completely integrable for transversely isotropic rods.

The magnetic rod system is generated by a Lax pair.

In the aligned force case, the system is superintegrable with purely periodic orbits.

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 non-canonical 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 Poincar\'e 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.