Integrability of an extensible conducting rod in a uniform magnetic field

G.H.M. van der Heijden & K. Yagasaki

The equilibrium equations for an isotropic Kirchhoff rod are known to be completely integrable. It is also known that neither the effects of extensibility and shearability nor the effects of a uniform magnetic field individually break integrability. Here we show, by means of a Melnikov-type analysis, that, when combined, these effects do break integrability giving rise to spatially chaotic configurations of the rod. A previous analysis of the problem suffered from the presence of an Euler-angle singularity. Our analysis provides an example of how in a system with such a singularity a Melnikov-type technique can be applied by introducing an artificial unfolding parameter. This technique can be applied to more general problems.

keywords: elastic rod, magnetostatics, Hamiltonian mechanics, integrability, homoclinic orbits, Melnikov analysis, horseshoe dynamics

Journal of Physics A: Mathematical and Theoretical 44, 495101 (2011)