# 4.2 Minimizing the energy of vector fields on surfaces of revolution  (Page 8/9)

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$E\left(\varphi \right)={\int }_{0}^{2\pi }\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\int }_{0}^{h}\left[\alpha ,\left(\varphi \left(\theta ,t\right),t\right),+,\beta ,\left(t\right),{\varphi }_{t},{\left(\theta ,t\right)}^{2},+,\gamma ,\left(t\right),{\varphi }_{\theta },{\left(\theta ,t\right)}^{2},+,\kappa ,\left(t\right)\right]dtd\theta$
$E\left(\stackrel{˜}{\varphi }\right)={\int }_{0}^{2\pi }\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\int }_{0}^{h}\left[\alpha ,\left(\varphi \left({\theta }_{0},t\right),t\right),+,\beta ,\left(t\right),{\varphi }_{t},{\left({\theta }_{0},t\right)}^{2},+,\kappa ,\left(t\right)\right]dtd\theta$

so that

$\begin{array}{cc}\hfill E\left(\varphi \right)-E\left(\stackrel{˜}{\varphi }\right)& ={\int }_{0}^{2\pi }{\int }_{0}^{h}\alpha \left(\varphi \left(\theta ,t\right),t\right)+\beta \left(t\right){\varphi }_{t}{\left(\theta ,t\right)}^{2}+\kappa \left(t\right)dt\hfill \\ \hfill -& {\int }_{0}^{h}\alpha \left(\varphi \left({\theta }_{0},t\right),t\right)+\beta \left(t\right){\varphi }_{t}{\left({\theta }_{0},t\right)}^{2}+\kappa \left(t\right)dt\hfill \\ \hfill +& {\int }_{0}^{h}\gamma \left(t\right){\varphi }_{\theta }{\left(\theta ,t\right)}^{2}dtd\theta \hfill \end{array}$

The expression

$\left({\int }_{0}^{h},\alpha ,\left(\varphi \left(\theta ,t\right),t\right),+,\beta ,\left(t\right),{\varphi }_{t},{\left(\theta ,t\right)}^{2},+,\kappa ,\left(t\right),d,t\right)$
$-\left({\int }_{0}^{h},\alpha ,\left(\varphi \left({\theta }_{0},t\right),t\right),+,\beta ,\left(t\right),{\varphi }_{t},{\left({\theta }_{0},t\right)}^{2},+,\kappa ,\left(t\right),d,t\right)$

is always nonnegative, since θ 0 is a minimum f . Similarly,

$\left({\int }_{0}^{h},\gamma ,\left(t\right),{\varphi }_{\theta },{\left(\theta ,t\right)}^{2},d,t\right)$

is positive, since $\gamma \left(t\right)$ is positive and ϕ θ is somewhere nonzero by assumption. Thus $E\left(\varphi \right)-E\left(\stackrel{˜}{\varphi }\right)>0$ . $\square$

## Non-uniqueness

In section 2.3, we prove that there is a unique $\varphi \left(\theta ,t\right):\left[0,2\pi \right]×\left[0,1\right]\to \mathbb{R}$ that is periodic in θ and satisfies $\Delta \varphi +\frac{sin\left(2\varphi \right)}{2}=0$ with given boundary conditions. (The proof actually holds for $t\in \left[0,h\right]$ , where $0 .) Furthermore, at least with horizontal boundary conditions, the short cylinder admits a vector field which attains the minimum energy. However, this is not the case on a sufficiently tall cylinder: $\varphi \left(\theta ,t\right)=0$ is an unstable critical function.

Given a cylinder of height h , consider ${\varphi }_{ϵ,h}\left(\theta ,t\right)=ϵt\left(h-t\right)$ . (Since this satisfies ${\varphi }_{ϵ,h}\left(\theta ,0\right)={\varphi }_{ϵ,h}\left(\theta ,h\right)=0$ , it describes a vector field with horizontal boundary conditions.) We can compute

$E\left({\varphi }_{ϵ,h}\right)={\int }_{0}^{2\pi }\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\int }_{0}^{h}{cos}^{2}\left(ϵ,t,\left(h,-,t\right)\right)+{\left(ϵh-2ϵt\right)}^{2}dtd\theta$
$=2\pi {\int }_{0}^{h}{cos}^{2}\left(ϵ,t,\left(h,-,t\right)\right)+{\left(ϵh-2ϵt\right)}^{2}dt$

Plotting $E\left({\varphi }_{ϵ,h}\right)-2\pi h$ with respect to ϵ and h yields the graph shown in the following figure:

There is a clear region on which $E\left(0\right)=2\pi h$ is greater than $E\left({\varphi }_{ϵ,h}\right)$ . It is simple to construct specific examples where ${\varphi }_{ϵ,h}$ has lower energy than the horizontal field; a more difficult task is to find the lowest h 0 that admits a lower-energy field. By Theorem 2, such an h 0 is bounded below by $\sqrt{8}$ . A series of examples bounds it above by $\sqrt{10}$ , but the possibility remains that a better bound is available.

Even without an exact value for h 0 , we may discuss the significance of its existence. On a cylinder less than a certain height, the horizontal vector field described by $\varphi \left(\theta ,t\right)=0$ uniquely minimizes energy; on any tall cylinder, a slight turn up or down will show an improvement. An energy-minimizing function on the tall cylinder, if it exists, must still satisfy the differential equation $\Delta \varphi +\frac{sin\left(2\varphi \right)}{2}=0$ , but $\varphi =0$ is a trivial solution to this. We are left with two possible situations.

Perhaps solutions $\varphi :\left[0,2\pi \right]×\left[0,h\right]\to \mathbb{R},\varphi \left(\theta ,0\right)=\varphi \left(\theta ,h\right)=0$ to the differential equation are unique for $0 and non-unique for ${h}_{0} . This conclusion is plausible, if slightly uncomfortable. Alternatively, it is not obvious that on a cylinder of height greater than h 0 , there exists a vector field that attains the minimal energy. The compactness properties of our space have not been adequately explored to say if such a situation makes sense.

## Computer approximations

We have seen that the partial differential equation $\Delta \varphi +\frac{sin\left(2\varphi \right)}{2}=0$ does not yield to any of the standard analytical solving techniques. This motivates us to seek out numerical methods to use a computer to approximate the solution. MATLAB is an excellent environment in which to pursue this goal, as it has a powerful fsolve command which rapidly and accurately solves partial differential equations.

## Polynomal interpolation and runge's phenomenon

When a computer “solves" a differential equation, it actually only assures that the differential equation is satisfied at a finite number of points. Between these points the computer uses polynomial interpolation to create a smooth solution. The question then arises as to how well we can trust the interpolation between these points. There is a classic example, called Runge's Phenomenon, of how interpolating using an evenly spaced grid leads to disastrous results. Suppose we would like to approximate the function

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