First integrals for elastic curves: twisting instabilities of helices

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