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

As we increase the degree of the approximation, and supposedly increase the accuracy of the approximation, the following figure is produced:

The red curve is the function we are trying to approximate on a set of evenly spaced points.The blue curve is a 5 ^th order approximation, and the green curve is a 9 ^th order approximation. As we can see, the approximation actually becomes worse; this is a pitfall we would like to avoid.

Chebyshev points, fourier matrices, and plots

To sidestep the possibility of suffering Runge's Phenomenon, we can redefine our grid using Chebyshev points instead of equally spaced points. The Chebyshev points cluster near the boundary of the grid (in the two dimensional case, the interval), and are more widely spaced in the center of the grid. This forces the interpolation to be much more accurate on the edges of the grid, bounding the error that can occur. In fact, using this method causes the interpolation to converge extremely rapidly.

To handle periodicity in the θ variable, we borrow some theory about Fourier Discretization Matrices from Trefethen's Spectral Methods in MATLAB [link] . These matrices allow us to solve our differential equation on the interior of the cylinder while leaving the boundary conditions fixed.

With this framework, our program allows us to input a height h , a radius function $r\left(t\right)$ , and two functions $f\left(\theta \right)=\varphi (\theta ,0)$ and $g\left(\theta \right)=\varphi (\theta ,h)$ that describe the boundary conditions, and finds a very close approximation of a function $\varphi (\theta ,t)$ which satisfies Equation [link] . Some images produced by this program can be seen in Section (10).

Limitations and future work

Decisions made at the beginning of our research period have, of course, led our work in a particular direction. As is often the case, these decisions placed certain limitations on our results that were not foreseen at the time. The limitations are not insurmountable, nor do they invalidate any of the above, but are worth thinking about.

We have primarily discussed a function $\varphi :[0,2\pi ]\times [0,h]\to \mathbb{R}$ , periodic in θ , as a representative for a vector field on a surface S . Certainly, every such function defines a vector field via the relation given in equation [link] . However, this vector field is not uniquely defined: ${\varphi }_n={\varphi }_0+2n\pi$ describes the same vector field as ϕ ₀ for all integers n .

Furthermore, not every vector field can be described by such a ϕ . By requiring ϕ to be periodic, the vector field $V\left(\varphi \right)$ must “untwist" as much as it “twists." This is convenient, for multiple reasons. Periodicity simplifies many of the integration by parts steps taken in our proofs. It also prevents a situation in which the top and bottom boundary conditions have different numbers of twists. Given such conditions, it is topologically impossible to define a continuous vector field between them. However, placing a periodicity requirement on ϕ also rules out perfectly valid vector fields: for example, a field with exactly one twist on every horizontal slice.

It would not be overly difficult, we imagine, to rethink our results to accommodate continuous vector vields that cannot be represented by periodic ϕ . One result which is slightly troublesome is uniqueness on a short cylinder (Theorem 1). Given boundary conditions that are constantly “straight up" on the bottom and “straight down" on the top, and a vector field V between them, one can construct a vector field $\tilde{V}$ with identical energy to V by having $\tilde{V}$ rotate clockwise wherever V rotates counterclockwise, and vice versa.

Perhaps another early limiting choice which was looking specifically at unit-length vector fields on unit-height cylinders with unit radius. Fixing three quantities greatly restricts the problem, and while this was certainly conductive to finding early results, we were misled to search for analogues in more general cases. Again, uniqueness comes to mind; somehow, the ratio between vector length and cylinder radius creates a special value between $\sqrt{8}$ and $\sqrt{10}$ . Once we turned our attention to surfaces of general radius and height, we did not find such a point, and it is quite possible that there is no equivalent.

There are a number of directions in which we can continue our research. Resolving the above questions is one such direction. We would also like to answer the questions about existence of solutions to Equation [link] that have nagged us since the beginning: is there always a vector field which attains the minimal energy value on any given surface? We have conjectured many “limiting results," as in "Limiting Results on the Cylinder" , that we suspect are true but have not proven yet.

Another problem that we have only just begun to examine is, in a way, a reversal of the current problem. Given a function $\varphi (\theta ,t):[0,2\pi ]\times [0,h]\to \mathbb{R}$ , can we find a radius function $r\left(t\right):[0,h]\to (0,\infty )$ describing a surface of rotation S such that the energy of the vector field described by ϕ on S is minimal over all surfaces of height h ? Since energy is inversely proportional to radius, $r\left(t\right)$ tends toward ∞ for all t ; we must somehow constrain r . Bounding its derivative and fixing the surface area of S are two approaches we have considered. This problem seems to have plenty of potential for future research.

Numerical plots

In this section we reproduce some numerical results as described in Section (7).

First, the plot of ϕ and surface for a cylinder of height 1 with boundary conditions $f\left(\theta \right)=cos\left(\theta \right)$ and $g\left(\theta \right)=sin\left(\theta \right)$

Second, the plot of ϕ and surface with radius $r\left(t\right)=2+sin\left(t\right)$ and boundary conditions $f\left(\theta \right)=cos\left(\theta \right)$ and $g\left(\theta \right)=sin\left(\theta \right)$

Summary

This report summarizes work done as part of the Calculus of Variations PFUG under Rice University's VIGRE program. VIGRE is a program of Vertically Integrated Grants for Research and Education in the Mathematical Sciences under the direction of the National Science Foundation. A PFUG is a group of Postdocs, Faculty, Undergraduates and Graduate students formed round the study of a common problem.

This module investigates the “kinetic energy" of unit-length vector fields on surfaces of rotation, focusing mainly on minimizing energy given a surface and boundary conditions. Questions of existence and uniqueness are explored.

Acknowledgements

This Connexions module describes work conducted as part of Rice University's VIGRE program, supported by National Science Foundation grant DMS-0739420. We would like to thank the faculty mentors, Bob Hardt, Leo Rosales, and Mike Wolf, and our graduate student mentors, Chris Davis and Renee Laverdiere, for all their help this summer. Without their assistance, this paper would be noticeably shorter. We are also indebted to a number of members of the math department who volunteered advice, notably Frank Jones and Rolf Ryham. Nearly all of the numerical approximation work is thanks to Mark Embree, who, after hearing us present our work, accomplished in an hour what we had been unable to do all summer. We also thank the undergraduate members of this group, Yan Digilov, Bill Eggert, Michael Jauch, Rob Lewis, and Hector Perez.