Classification of the Spatial Equilibria of the Clamped Elastica: Symmetries and Zoology of Solutions

We investigate the congurations of twisted elastic rods under applied end loads and
clamped boundary conditions. We classify all the possible equilibrium states of inextensible, unshearable, isotropic, uniform and naturally straight and prismatic rods. The Kirchhofi equations which describe the equilibria of these rods are integrated in a formal way which enable us to describe the boundary conditions in terms of 2 closed form equations involving 4 free parameters. We show that only reversible solutions of the Kirchho equations fulfill the clamped boundary conditions and we sort them according to their period in the phase plane. We show how planar untwisted configurations as well as circularly closed configurations play an important role in the classification.