On the stability of rotating heavy chains

The object of this paper is two-fold. On the one hand we want to develop the duality theory for critical points which is introduced in [4]. The main result in this context is that if two critical points are in duality, and one of them is a local minimiser of its functional, then so is the other. In principle then it is possible to draw conclusions about the stability of stationary solutions of a system, not by analysing the potential energy functional, but by examining an appropriate functional which is dual to it. Our second aim is to illustrate this idea by means of the specific example of the spinning chain problem. It turns out that in this case the dual variational problem is much more tractable than the potential energy functional, and indeed the stability analysis follows at once from results in the literature, once the duality is taken into account.