Nonlinear dynamics of filaments I. Dynamical instabilities

The Kirchhoff model provides a well-established mathematical framework to study, both computationaly and theoretically, the dynamics of thin filaments within the approximations of linear elasticity theory. The study of static solutions to these equations has a long history and the usual approach to describing their instabilities is to study the time-dependent version of the Kirchhoff model in the Euler angle frame. Here we study the linear stability of the full, time-independent, equations by introducing a new arc length preserving perturbation scheme. As an application, we consider the instabilities of various stationary solutions, such as the planar ring and straight rod, subjected to twisting perturbations. This scheme gives a direct proof of the existence of dynamical instabilities and provides the selection mechanism for the shape of unstable filaments.