4.3 Minimizing the energy of vector fields on surfaces of revolution

Introduction

It is highly recommended that the reader familiarize themselves with [2] before approaching this module. Both the problems and the techniques used to solve them will draw heavily from that work. Wewill first answer questions previously raised concerning the unit cylinder before moving on to the study of similar vector fields onthe torus.

The cylinder

First, we turn our attention to the unit cylinder (with radius and height 1) for futher investigation on [2] work.

Parametrization and equations

We borrow heavily from [2] . Our surface, the unit cylinder with radius $r=1$ may be parameterized as $\Phi (\theta ,t)=(cos\theta ,sin\theta ,t)$ . Our vector field is parameterized as $V\left(\varphi \right)=(-sin\theta cos\varphi ,cos\theta cos\varphi ,sin\varphi ):0\le \theta \le 2\pi ,0\le t\le 1$ and the corresponding energy equation on the cylinder is

which, of course, is a specific case of the general equation

As [2] shows, the minimizing $\varphi$ must satisfy

$\theta$ -independence

We recall and restate Theorem 5 from [2] :

Let a surface $S$ with radius $r\left(t\right)$ be given. Suppose that the boundary conditions $\varphi (\theta ,0),\varphi (\theta ,h)$ of a vector field on $S$ do not depend on $\theta$ : that is, $\varphi (\theta ,0)={\varphi }_0,\varphi (\theta ,h)={\varphi }_h$ for all $\theta \in [0,2\pi ]$ and constant ${\varphi }_0,{\varphi }_h$ . The function $\varphi (\theta ,t)$ which minimizes energy given constant boundary conditions $\varphi (\theta ,0)={\varphi }_0,\varphi (\theta ,h)={\varphi }_h$ does not depend on $\theta$ . In other words, the vector field described by $\varphi$ is constant along every ”horizontal slice" of the surface.

Existence and uniqueness

We now aim to show that such a unique energy-minimizing $\varphi$ does in fact exist on this cylinder.

Theorem 1

Let $(-sin\theta cos{\varphi }_0,cos\theta cos{\varphi }_0,sin{\varphi }_0)$ and $(-sin\theta cos{\varphi }_1,cos\theta cos{\varphi }_1,sin{\varphi }_1)$ be “constant” boundary data on the bottom and top of the unit cylinder respectively. Then the minimizer exists.

Proof:

Let $E={inf}_{\left\{Vtangentsmoothunitvectorfield\right\}}E\left(V\right)$ . Let $\mathcal{C}$ be the following the set of smooth functions:

Lemma 1

Suppose that $V_k$ are a sequence of unit tangent vector fields with $E\left(V_k\right)\to E$ . Then we can replace this sequence by another sequence of vector fields $V_k$ so that when we write the $V_k$ using an angle fucntion ${\varphi }_k$ , ${\varphi }_k$ is constant in $\theta$ and so that the energies $E\left(V_k\right)$ still approach $E$ .

Proof:

Suppose that $V_k$ depends on $\theta$ . Then there exists a $\theta$ -independent $V_j$ such that $E\left(V_j\right)\le E\left(V_k\right)$ . $E\left(V_k\right)\to E$ thus implies $E\left(V_j\right)\to E$ .