  We put forward a variational framework suitable for the study of curves whose
energies depend on their bend and twist degrees of freedom. By employing the
material curvatures to describe such elastic deformation modes, we derive the
equilibrium equations representing the balance of forces and torques on the
curve. The conservation laws of the force and torque on the curve, stemming
from the Euclidean invariance of the energy, allow us to obtain first integrals
of the equilibrium equations. To illustrate this framework, we apply it to
determine the first integrals for isotropic and anisotropic Kirchhoff elastic
rods, whose energies are quadratic in the material curvatures. We use them to
analyze perturbatively the deformations of helices resulting from their
twisting. We examine three kinds of twisting instabilities on unstretchable
helices, characterized by their wavenumbers, depending on whether their
boundaries are fixed, displaced along the radial direction or orthogonally to
it. We also analyze perturbatively the effect of the bending anisotropy on the
deformed states, which introduces a coupling between deformation modes with
different wavenumbers.
