4.2 Minimizing the energy of vector fields on surfaces of revolution  (Page 7/9)

On Γ ₁ , ${cos}^2\left(\varphi \right)\ge {cos}^2\left(m\right)$ , with strict inequality when $\varphi (\theta ,t)<m$ . Then by a similar argument as in Theorem 3, we can find a smooth function which approximates $\tilde{\varphi }$ with an energy that converges to that of $\tilde{\varphi }$ . Since, by assumption, there are some $(\theta ,t)$ where this occurs, we can conclude that $E\left(\varphi \right)>E\left(\tilde{\varphi }\right)$ and thus ϕ is not a minimizer. $\square$

Some results on general surfaces

Until now, most of our work has focused on the cylinder of unit radius. Working on an arbitrary surface of revolution requires much more complicated equations, but much of the theory is the same.

Let our surface S be parametrized by $\Phi (\theta ,t)=\left(r,\left(,t,\right),cos,\left(,\theta ,\right),,,r,\left(,t,\right),sin,\left(,\theta ,\right),,,t\right):0\le \theta \le 2\pi ,0\le t\le h,$ where $r\left(t\right)$ is the radius of S at height t . We would again like to define a vector field tangent to S by rotating the horizontal tangent vector field $(-sin(\theta ),cos(\theta ),0)$ by an angle $\varphi (\theta ,t)$ . This rotation must take place inside the tangent plane- in other words, we want to rotate $(-sin(\theta ),cos(\theta ),0)$ by $\varphi (\theta ,t)$ around $\frac{\partial \Phi }{\partial \theta }\times \frac{\partial \Phi }{\partial t}$ . The resulting vector field $V(\theta ,t)$ takes the form

The energy of the vector field defined by ϕ is given by

where

Note that $\alpha ,\beta ,\gamma ,\kappa$ are all nonnegative, and that this energy agrees with equation [link] when $r\left(t\right)=1$ .

The euler-lagrange equation

Using the methods of section 2.2, we can calculate the Euler-Lagrange equation of energy on a general surface:

where

Again, note that this agrees with equation [link] when $r\left(t\right)=1$ . This equation is not particularly enlightening in this form; without a given $r\left(t\right)$ , we can say very little, and even with a known $r\left(t\right)$ , it is no easier to solve than equation [link] . Any $\varphi (\theta ,t)$ which describes an energy-minimizing vector field on a given surface will satisfy equation [link] . This PDE also can be put to great use in our numerical approximations.

θ -independent vector fields

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 θ : that is, $\varphi (\theta ,0)={\varphi }_0,\varphi (\theta ,h)={\varphi }_h$ for all $\theta \in [0,2\pi ]$ and constant ${\varphi }_0,{\varphi }_h$ .

Theorem 5. 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 θ . In other words, the vector field described by ϕ is constant along every horizontal “slice" of the surface.

Proof: Let a θ -dependent $\varphi (\theta ,t)$ be given. We will find a θ -independent $\tilde{\varphi }(\theta ,t)$ with lower energy.

Consider the function $f\left(\theta \right)={\int }_0^h\alpha (\varphi (\theta ,t),t)+\beta \left(t\right){\varphi }_t{(\theta ,t)}^2+\kappa \left(t\right)dt$ for $\theta \in [0,2\pi ]$ . Since the domain is compact, there is some θ ₀ which minimizes f . Define $\tilde{\varphi }(\theta ,t)=\varphi ({\theta }_0,t)$ . Essentially, $\tilde{\varphi }$ takes the vertical slice of the vector field described by ϕ with minimal “vertical energy," and pastes that slice all around the surface. We compute: