4.2 Minimizing the energy of vector fields on surfaces of revolution

Introduction

In his classic 1710 treatise, the Théodicée , philosopher and mathematician Gottfried Leibniz set out to explore the problem of evil. Faced with the question of how an omniscient and completely benevolent deity could create a world in which pain and misfortune were ever-present, Leibniz proposed the theory of the “best of all possible worlds." Our world, he argued, must have been chosen by God because it maximizes “goodness" overall. Were anything different, even just slightly– say, the death toll of World War II were lower– then in the grand scheme of things, the world would actually be worse.

Leibniz's argument foresaw, at least philosophically, the development of the calculus of variations in mathematics. One can associate a “cost" with a “path" between two “points," which depends on how the path changes in time and space. Here the words “path," “point," and “cost" are used very abstractly. In the Théodicée , Leibniz took as a “path" the sequence of events in the world, between the beginning and end of time; the “cost" of this path is its “goodness." (Defining “goodness," of course, is another problem in itself.) Calculus of variations aims to determine, given such a cost, which path will minimize it. Seen from this perspective, Leibniz imagined God as the ultimate mathematician.

This summer, our VIGRE group has spent eight weeks applying techniques from the calculus of variations to a problem similar to Leibniz's, albeit of slightly reduced scope. On a surface of rotation (imagine a shape made with clay on a pottery wheel), one can define a unit-length vector field (imagine an infinite collection of arrows, all of equal length, such that one arrow is tacked to every point on the shape). There is a sense of “energy" to this vector field; a field in which every arrow points in the same direction is boring compared to one in which the arrows spin wildly. We take this “energy" to be our cost; our paths are the possible vector fields that can be placed on the surface. Our goal is to determine, given a specific shape and a vector field on its boundaries, what vector field on the rest of the shape has minimal energy. In examining this and related questions, we have touched on topics from a number of fields, including functional analysis, differential geometry, and topology.