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

Abstract

Abstract. The equilibrium equations for the isotropic Kirchhoff rod are known to form an integrable system. It is also known that the effects of extensibility and shearability of the rod do not break the integrable structure. Nor, as we have shown in a previous paper does the effect of a magnetic field on a conducting rod. Here we show, by means of Mel'nikov analysis, that, remarkably, the combined effects do destroy integrability; that is, the governing equations for an extensible current-carrying rod in a uniform magnetic field are nonintegrable. This result has implications for possible configurations of electrodynamic space tethers and may be relevant for electromechanical devices.

Nonintegrability of an extensible conducting rod in a uniform magnetic field

“…The equilibrium equations for an extensible and shearable rod in a uniform magnetic field were derived in [17]. The equations were obtained by reduction of a nine-dimensional system in Lie-Poisson form with three Casimirs to a six-dimensional canonical Hamiltonian system in terms of three Euler angles (θ, ψ, φ) and their canonical momenta (p θ , p ψ , p φ ), similar to the reduction of the isotropic elastic rod [20] or the symmetric heavy top [5].…”

“…where T u > 0 is a constant, L s is a 2 × 4 matrix consisting of row eigenvectors corresponding to eigenvalues with negative real parts of the Jacobian matrix J D 2 H(ξ 0 ) and ξ u 0 is a point on W u (ξ 0 ). Thus, we solve the boundary value problem (15) and (17) and continue the solution in ξ u 0 to compute W u (ξ 0 ) numerically. Similarly, to compute W s (ξ 0 ), we solve the boundary value problem (15) and…”

“…The problem was studied by one of us in a previous paper [17], where a two-degreesof-freedom system was obtained by reducing the rod equations in Lie-Poisson form to a canonical Hamiltonian system in terms of Euler angles and their canonical momenta. Numerical evidence of spatially complex rod configurations was given in the form of chaotic Poincaré sections.…”

“…Here we use a different method that removes the singularity from the unperturbed equilibrium with homoclinic orbits by introducing an artificial unfolding that corresponds to a change of coordinates. By a careful scaling of the equations, the magnetic rod is regarded as the unperturbed problem and extensibility/shearability is viewed as the perturbation (the opposite to the scaling used in [17]). We then apply a version of Melnikov's method due to Lerman and Umanskii [10] and Wiggins [22], briefly reviewed in the appendix for the reader's convenience, to show that there exist transverse homoclinic orbits to a saddle-focus.…”

Shape of an elastica under growth restricted by friction

“…In classical problems of elastic stability (like Euler buckling), the system is subject to conservative loads and stability of a static solution against infinitesimally small perturbations depends on second variations of an energy functional. Non-conservative forces like a follower load (Langthjem and Sugiyama, 2000), magnetic forces (Sinden and van der Heijden, 2009) or sliding friction can be treated either via an extended energy approach or by direct analysis of the equations of motion. The loss of elastic stability plays a key role in many processes related to biological growth, which is subject of the theory of morpho-elasticity (Goriely, 2017).…”