Relaxation Under Topological Constraints

Abstract

This contribution provides an informal introduction to the technique of magnetic relaxation, whereby an extremely wide family of solutions of the equations of magnetostatics, and of analogous steady solutions of the Euler equations, may be obtained, and their stability investigated. We approach this problem through the simpler, and physically more transparent, problem of gravitational relaxation of an incompressible medium of non-uniform density. We then describe the magnetic relaxation technique which yields solutions of nontrivial field topology, and we discuss the contrasting stability criteria for these magnetostatic states and for the analogous Euler flows. Applications to the theory of vortons (i.e, blobs of propagating vorticity) and to the problem of determining ‘energy’ invariants of knots and links are then discussed. The chapter concludes with a discussion of alternative relaxation procedures involving artificial modification of the Euler equations in a manner that conserves vorticity topology.