Stabilityofdeformations of Elastic Rods

Mechanics is a fascinating field of physics. Its main purpose is to study the interaction of particles in the presence of forces such as gravity. In contrast to the dynamics of particles, the study of continuous media, such as elastic bodies, is challenging and still today not entirely well-understood. This is due to the fact that continuous media behave in numerous and complex ways. For example it is difficult to describe mathematically the twisting and bending of elastic rods but approximate models do exist. The study of the dynamics of elastic rods is a field of research that is interesting in itself because it gives rise to beautiful and complex mathematical structures. Furthermore, understanding elastic rods is crucial for applications in several domains of science. In this proposal I will focus on a model that has been proven to be quite successful in describing the dynamics of rods in a certain approximation: the set of Kirchhoff equations. These equations are differential equations, that is they are equations involving derivatives. The solutions to these equations represent possible behaviors for elastic rods. The goal of the proposal is to study two types of solutions to the Kirchhoff equations: the periodic and soliton solutions. These two types of solutions have been proven to be very useful in other fields of science such as optics and fluid mechanics. More precisely, the goal of the proposal is to study the stability properties of solutions of the Kirchhoff equations. Stability is a fundamental concept in physics. An illustration of this concept is given by trying to make a pencil stand on its lead. In theory, it is possible but, in practice, because it is such an unstable state, it cannot be done. The same concept applies to solutions of differential equations: some are stable and some are not. In the case of differential equations though, it is often necessary to develop some sophisticated mathematical methods to study stability. However, it is crucial to study the stability of the solutions of the Kirchhoff equations because only stable solutions can be observed experimentally. The research of this proposal can be summarized as followed: • Develop a method to study the stability of certain solutions in the context of elasticity. • Apply this method to the equations I am studying to obtain the stable solutions. • Study the physical conditions under which these stable solutions propagate in a rod.