The classical problem of the buckling of an elastic rod in a magnetic ¯eld is investigated
using modern techniques from dynamical systems theory. The Kirchhoff equations,
which describe the static equilibrium equations of a geometrically exact rod under end
tension and moment are extended by incorporating the evolution of a fixed external
vector (in the direction of the magnetic field) that interacts with the rod via a Lorentz
force. The static equilibrium equations (in body cordinates) are found to be noncanonical
Hamiltonian equations. The Poisson bracket is generalised and the equilibrium equations
found to sit, as the third member, in a family of rod equations in generalised magnetic
fields. When the rod is linearly elastic, isotropic, inextensible and unshearable the equations
are completely integrable and can be generated by a Lax pair.
The isotropic system is reduced using the Casimirs, via the Euler angles, to a four-dimensional
canonical system with a first integral provided the magnetic field is not
aligned with the force within the rod at any point as the system losses rank. An energy
surface is specified, defning three-dimensional flows. Poincare sections then show closed
curves.
Through Mel'nikov analysis it is shown that for an extensible rod the presence of a
magnetic field leads to the transverse intersection of the stable and unstable manifolds
and the loss of complete integrability. Consequently, the system admits spatially chaotic
solutions and a multiplicity of multimodal homoclinic solutions exist. Poincare sections
associated with the loss of integrability are displayed.
Homoclinic solutions are computed and post-buckling paths found using continutaion
methods. The rods buckle in a Hamiltonian-Hopf bifurcation about a periodic
solution. A codimension-two point, which describes a double Hamiltonian-Hopf bifurcation,
determines whether straight rods buckle into localised configurations at either two
critical values of the magnetic field, a single critical value or do not buckle at all. The
codimension-two point is found to be an organising centre for primary and multimodal
solutions