Bifurcation theory of an elastic conducting wire subject to magnetic forces

In this paper we study the equilibrium states of a nonlinearly elastic wire in a magnetic field. The wire is perfectly flexible, is suspended between fixed supports and carries an electric current. We consider two problems. The first in which the magnetic field is constant can be solved exactly. The set of solutions illustrates the phenomenon of symmetry breaking which is a chapter in the theory of imperfect bifurcation. The second problem is one in which the magnetic field is produced by current flowing in a pair of infinitely long parallel wires. When the line of supports of the elastic wire is parallel to these and equidistant from them we may apply the global bifurcation results of Crandall and Rabinowitz to study the set of solutions. We also consider perturbations of this case. This is another example of imperfect bifurcation.