Nonlinear Contour Dynamics in Macroscopic Systems

Abstract

In this chapter we study several macroscopic applications of the closed contour dynamics problem by using theorems for differential geometry. A first application presented is the study of the geometry of trajectories of charged particles in magnetic fields. We present some closeness trajectories criteria based on Bonnet and Fenchel theorems. Another example is given by the application of the Gauss–Bonnet theorem to problems of trapping particles inside closed magnetic surfaces. At larger physical scales, we present the occurrence of very localized stable waves orbiting around elastic spheres, and we conclude the chapter with a description of nonlinear modes in neutron stars.