## Integrability of a conducting elastic rod in a magnetic field

D. Sinden & G.H.M. van der Heijden

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 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.
keywords: elastic rod, magnetic field, non-canonical Hamiltonian system,
integrability, superintegrability, Casimir invariants, Lax pair

*Journal of Physics A: Mathematical and Theoretical* **41**,
045207 (2008)