2007/08/08 by D. Sinden, David Sinden, G.H.M. van der Heijden +1 · 8 citations
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Condensed matter physics #Elasticity and Wave Propagation #Field (mathematics) #Geotechnical and Geomechanical Engineering #Magnetic field #Mathematics #Physics #Pure mathematics #Quantum mechanics #math-ph #math.MP #msc:70H06 #msc:74K10 #msc:78A30
paper · pdf · doi:10.1088/1751-8113/41/4/045207
published in Journal of Physics A Mathematical and Theoretical 41(4), 045207 (Institute of Physics) · 19 pages, 1 figure
arxiv created 2007/08/08 · openalex publication_date 2008/01/15 · arxiv updated 2010/07/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 family 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.