4.3 Minimizing the energy of vector fields on surfaces of revolution  (Page 4/4)

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$t$ -independence

Consider a $\varphi$ that depends solely on $s$ and is thus independent of $t$ . In this case, the Euler-Lagrange equation is

$\alpha \left(R,r,t\right){\varphi }^{\text{'}\text{'}}\left(s\right)+\kappa \left(R,r,t\right)sin\left(2\varphi \left(s\right)\right)=0$
$\alpha {\varphi }^{\text{'}\text{'}}{\varphi }^{\text{'}}+\kappa {\varphi }^{\text{'}}sin\left(2\varphi \right)=0$
${\left(\frac{\alpha }{2}{\left({\varphi }^{\text{'}}\right)}^{2}-\frac{\kappa }{2}cos\left(2\varphi \right)\right)}^{\text{'}}=0$
$\alpha {\left({\varphi }^{\text{'}}\right)}^{2}-\kappa cos\left(2\varphi \right)=2c$
${\varphi }^{\text{'}}=\sqrt{\frac{\kappa }{\alpha }cos\left(2\varphi \right)+{c}_{1}\left(t\right)},\phantom{\rule{0.166667em}{0ex}}{c}_{1}=\frac{2c}{\alpha }$
$\int \frac{d\varphi }{\sqrt{{c}_{1}+\frac{\kappa }{\alpha }cos\left(2\varphi \right)}}=\int ds.$

Thus, as in the case of the cylinder, we compute an explicit expression for $\varphi$ .

Special minimizers

However, we note that the minimizing $\varphi$ must independent both of $s$ and $t$ . When $\varphi$ is constant, ${\varphi }_{t}={\varphi }_{s}\equiv 0$ . In this case, the Euler-Lagrange equation is condensed to

$\frac{R\left(R+2rcos\left(t\right)\right)}{r\left(R+rcos\left(t\right)\right)}sin\left(2\varphi \right)=0$

which implies that, since $R>r>0$ , $sin2\varphi =0$ . This in turn implies that the minimizing $\varphi =0$ or $\frac{\pi }{2}$ .

We note that $E\left(0\right)=\frac{4{\pi }^{2}r}{\sqrt{{R}^{2}-{r}^{2}}}$ and $E\left(\frac{\pi }{2}\right)=\frac{-4{\pi }^{2}\left(-2R+\sqrt{{R}^{2}-{r}^{2}}\right)}{r}$ . Thus, the relationship between $E\left(0\right)$ and $E\left(\frac{\pi }{2}\right)$ depends upon the relationship between $r$ and $R$ .

The inequality $E\left(0\right) , written equivalently

$\frac{4{\pi }^{2}r}{\sqrt{{R}^{2}-{r}^{2}}}<\frac{-4{\pi }^{2}\left(-2R+\sqrt{{R}^{2}-{r}^{2}}\right)}{r},$

is true when $r<\frac{\sqrt{3}}{2}R . In turn, $E\left(0\right)>E\left(\frac{\pi }{2}\right)$ when $\frac{\sqrt{3}}{2}R . Notice $\int \stackrel{˜}{c}dt=\frac{\pi r\left(-2+\frac{R}{\sqrt{{R}^{2}-{r}^{2}}}\right)cos\left(2\varphi \right)}{r}$ (when $\varphi$ is independent of $s,t$ ) vanishes when $r=\frac{\sqrt{3}}{2}R$ . This implies that when $r=\frac{\sqrt{3}}{2}R$ , any constant $\varphi$ has equal energy .

We may continue this investigation of the effect of the torus' radii upon the associated energy. We may choose a few promising limits andsurvey the resulting limits of energy.

 $E\left(0\right)$ $E\left(\frac{\pi }{2}\right)$ $R\to \infty$ 0 $\infty$ $r\to R$ $\infty$ $8{\pi }^{2}$ $r\to 0$ 0 $\infty$

$s$ -independence

Let a torus $\Phi \left(s,t\right)=\left(cos\left(s\right)\left(R+rcos\left(t\right)\right),sin\left(s\right)\left(R+rcos\left(t\right)\right),rsin\left(t\right)\right),$ with $0\le s,r\le 2\pi ,\phantom{\rule{0.166667em}{0ex}}R>r>0$ be given.

Theorem 2

The function $\varphi \left(s,t\right)$ which minimizes energy does not depend on $s$ .

Proof:

Let

$g\left(s\right)={\int }_{0}^{2\pi }a\left(R,r,t\right){\varphi }_{t}^{2}+c\left(R,r,t\right)+\stackrel{˜}{c}\left(R,r,t,\varphi \left(s,t\right)\right)dt$

$s\in \left[0,2\pi \right]$

By compactness, $g\left(s\right)$ assumes a minimum at some $s={s}_{0}$ .

$E\left(\varphi \left(s,t\right)\right)-E\left(\varphi \left({s}_{0},t\right)\right)=$
${\int }_{0}^{2\pi }\left[\left({\int }_{0}^{2\pi },a,{\left(R,r,t\right)}^{2},{\varphi }_{t},{\left(s,t\right)}^{2},+,c,\left(R,r,t\right),+,\stackrel{˜}{c},\left(R,r,t,\varphi \left(s,t\right)\right),d,t\right)$
$-\left({\int }_{0}^{2\pi },a,{\left(R,r,t\right)}^{2},{\varphi }_{t},{\left({s}_{0},t\right)}^{2},+,c,\left(R,r,t\right),+,\stackrel{˜}{c},\left(R,r,t,\varphi \left({s}_{0},t\right)\right),d,t\right)$
$+\left({\int }_{0}^{2\pi },b,\left(R,r,t\right),{\varphi }_{s},{\left(s,t\right)}^{2},d,t\right)\right]ds>0.$

Therefore, $s$ -independent unit vector fields have a lower energy on the torus than any unit vector field that depends on $s$ .

Second variation

We now examine the stability of the proposed minima $\varphi =0$ and $\varphi =\frac{\pi }{2}$ . We do so by examining the second variation by way of the Euler-Lagrange equation.

${J}^{\text{'}\text{'}}\left(0\right)={\int }_{0}^{2\pi }{\int }_{0}^{2\pi }\frac{2}{r\left(R+rcos\left(t\right)\right)}\left[{\eta }^{2}cos2\varphi \left({R}^{2}+2Rrcos\left(t\right)\right)$
$+{\eta }_{t}^{2}\left(R+rcos{\left(t\right)}^{2}\right)+{\eta }_{s}^{2}{r}^{2}\right]\phantom{\rule{4pt}{0ex}}dsdt$

$\varphi \equiv 0$ is clearly a stable minimum when $R\ge 2r$ . When $\int \int \frac{2}{r\left(R+rcos\left(t\right)\right)}\left[{\eta }^{2}\left({R}^{2}+2rRcos\left(t\right)\right]<\frac{2}{r\left(R+rcos\left(t\right)\right)}\left[{\eta }_{t}^{2}{\left(R+rcos\left(t\right)\right)}^{2}+{\eta }_{s}^{2}{r}^{2}\right]$ , $\varphi =\frac{\pi }{2}$ is a stable minimum. More investigation on this last condition is a likely candidate for future work.

General surfaces of revolution

Although we focused our work largely on the torus and cylinder, we also began treating general surfaces of revolution. In doing so, weencountered several problems. First, we notice that any surface parametrized by $\Phi \left(x,y\right)=\left(x,y,g\left(x,y\right)\right)$ is difficult to work with due to the fact that using the Gram-Schmidt method to arrive at an expressionfor energy becomes unwieldy.

A parametrization $\Phi \left(x,y\right)=\left(x,y,g\left(y\right)\right)$ is much easier to work with ( ${\Phi }_{x}$ and ${\Phi }_{y}$ are, in fact, orthonormal), but defining boundary condtions becomes difficult. Parametrizing the surface as

$\Phi \left(r,\theta \right)=\left(rcos\theta ,rsin\theta ,f\left(r\right)\right)$

(as [2] does) greatly simplifies thinking about boundary conditions; they can be defined similarly to those on the unit cylinder. We mustkeep in mind, though, that $r>0$ to avoid a singularity at the origin.

In the case of this last parametrization, we may hope to arrive at a proof that, for any boundary conditions, there exists a continuoustangent unit vector field on $\Phi$ . A fruitful approach might involve allowing $\varphi$ to simply change linearly; we leave the details for the future.

Other future work

The most pressing piece of future work is the proof of the existence of a minimizer $\varphi$ on the torus. It is our hope that the method that we used for the analogous proof on the cylinder might be adaptableto the torus; however, it seems that there are some fundamental differences between the two $\varphi$ functions.

As was mentioned earlier, more investigation into the stability of minima on the torus is required: we hope to further explore the natureof $\varphi =\frac{\pi }{2}$ on the torus.

Acknowledgments

We would like to thank the Rice VIGRE program and the National Science Foundation for making our research possible. We would also like tothank our faculty and graduate advisors, Leobardo Rosales, Robert Hardt, Michael Wolf, Ryan Scott, and Evelyn Lamb, as well as the entireRice University Department of Mathematics. We also thank our undergraduate students James Hart, Conor Loftis, Aneesh Mehta, and Anand Shah.

This research was supported by National Science Foundation grant DMS-0739420

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