Solving Surface Optimization Problem w/Calculus of Variations

In summary: This is because as the difference between the radii of the two hoops becomes larger, the soap film will continue to stretch farther and the surface area will continue to increase.
  • #1
TriTertButoxy
194
0
Ok, so here's the problem:

Two circular wire hoops of radius [itex]R[/itex] are spaced a distance [itex]2\ell[/itex] apart. Consider a soap film stretching beatween the two hoops. Due to surface tension the film's equlibrium form is a surface of minimal area. Use the calculus of variations to find this minimal surface.

What happens when [itex]\ell/R[/itex] is small? What happens when [itex]\ell/R[/itex] becomes large? Is there a critical value of [itex]\ell/R[/itex]? If so, explain why; find in numerically; and say what happens when one exceeds it.

Calculus of Variations is simple: To "optimize" a quantity, [itex]y(x)[/itex], (i.e. find its minima/maxima), you find the value of [itex]x[/itex] which satisfies [itex]y'(x)=0[/itex].

If the independent is a function which optimizes a quantity (as in the case of this problem),
[tex]A=\int F(y(x),\,y'(x);\,x)\,dx[/tex]​
the optimized [itex]y(x)[/itex] is given by the solution to the differential equation below (Euler's Equation):
[tex]\frac{\partial F}{\partial y}-\frac{d}{dx}\left(\frac{\partial F}{\partial y'}\right)=0[/tex]​


Here's what I did. It makes most sense to set this problem in cylindrical coordinates. So, the surface area of the soap film is given by

[tex]A=\int_0^{2\ell}2\pi r(z)\sqrt{1+r'(z)^2}dz.[/tex]​

Here, we are assuming that there is no angular dependence of [itex]r[/itex]. So in anticipation of applying the calculus of variations (Euler's Equation), I evaluated the following quantities for [itex]F(r(z),\,r'(z);\,z)=r(z)\sqrt{1+r'(z)^2}[/itex]:

[tex]\frac{\partial F}{\partial r}=\sqrt{1+r'^2}\,,\,\,\,\,\,\,\,\,\frac{\partial F}{\partial r'}=\frac{r(z)r'(z)}{\sqrt{1-r'^2}}\,,\,\,\,\,\text{and}\,\,\,\,\,\,\frac{d}{dz}\left(\frac{\partial F}{\partial r'}\right)=\frac{1-r'^4+rr''}{(1+r'^2)^{3/2}}[/tex]​

Therefore Euler's equation becomes,
[tex]\frac{1+r'^2-rr''}{(1+r'^2)^{3/2}}=0.[/tex]​
If we disallow the denominator from vanishing, we can simplify this:
[tex]1+\left(\frac{dr}{dz}\right)^2-r(z)\frac{d^2r}{dz^2}=0.[/tex]​

I have no clue how to approach this. Mathematica gave me some really weird ugly answer, which when I asked it to FullSimplify, gave an unmanagable expression with hyperbolic sines and cosines. What's the solution to this differential equation, and how do I analytically apply the boundary conditions?
 
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  • #2
The solution to the differential equation is r(z) = A \cosh \left(\frac{z-B}{A}\right),where A and B are constants. To apply the boundary conditions, we need to know what the radii of the two hoops are. Let the radius of the first hoop be R_1 and the radius of the second hoop be R_2. Then we can solve for A and B in terms of R_1 and R_2:A = \frac{1}{2} \sqrt{R_2^2 - R_1^2}, B = \ell + \frac{1}{2} \log\left(\frac{R_2 + R_1}{R_2 - R_1}\right).What happens when \ell/R is small? When \ell/R is small, the surface area of the soap film will be small as well. This is because the difference between the radii of the two hoops is small, so the soap film will not stretch very far and thus the surface area will be small.What happens when \ell/R becomes large? When \ell/R becomes large, the surface area of the soap film will increase. This is because the difference between the radii of the two hoops is large, so the soap film will stretch farther and thus the surface area will be larger.Is there a critical value of \ell/R? If so, explain why; find in numerically; and say what happens when one exceeds it.No, there is no critical value of \ell/R. The surface area of the soap film will continue to increase as \ell/R increases without bound.
 
  • #3


I find this problem very interesting and challenging. The problem of finding the minimal surface area of a soap film between two circular hoops can be solved using the calculus of variations. This method allows us to optimize a quantity by finding the value of the independent variable that satisfies a certain condition. In this case, we want to minimize the surface area of the soap film.

When \ell/R is small, we can assume that the soap film is almost flat, and the problem becomes similar to finding the minimal surface area of a rectangular sheet stretched between two parallel lines. In this case, the minimal surface area would be a plane. As \ell/R increases, the film becomes more curved and the minimal surface area is no longer a plane. Instead, it takes on a more complex shape.

As \ell/R becomes larger, the soap film becomes more stretched and the surface tension becomes more dominant. This results in a smaller surface area as the film tries to minimize its energy. There is a critical value of \ell/R where the film can no longer support its own weight and collapses, resulting in a singularity in the solution. This critical value can be found numerically by solving the differential equation and applying the boundary conditions.

In order to solve the differential equation, we can use numerical or analytical methods. Numerical methods involve approximating the solution using a computer, while analytical methods involve finding a closed-form solution. In this case, it may be difficult to find an analytical solution due to the complexity of the differential equation.

To apply the boundary conditions, we need to consider the geometry of the problem. The two circular hoops provide the boundaries for the soap film, and we can assume that the film is continuous at these boundaries. We can also consider the symmetry of the problem, where the film is symmetric about the midline between the two hoops. These boundary conditions can be used to find the optimal solution for the minimal surface area.

In conclusion, using the calculus of variations, we can find the minimal surface area of a soap film between two circular hoops. As \ell/R increases, the film becomes more curved and the minimal surface area decreases. There is a critical value of \ell/R where the film collapses, resulting in a singularity in the solution. Further research and analysis can be done to explore the behavior of the soap film as \ell/R exceeds the critical value.
 

1. What is the calculus of variations?

The calculus of variations is a branch of mathematics that deals with finding the optimal function or curve that minimizes or maximizes a given functional. It is used to solve problems involving optimization, such as finding the shortest path between two points or the shape of a soap bubble.

2. How is the calculus of variations applied to surface optimization problems?

In surface optimization problems, the goal is to find the shape of a surface that minimizes or maximizes a given functional, such as minimizing surface area or maximizing volume. The calculus of variations is used to find the critical points of the functional, which correspond to the optimal surface shape.

3. What are the steps involved in solving a surface optimization problem with calculus of variations?

The first step is to define the functional that represents the surface optimization problem. Then, the Euler-Lagrange equation is used to find the critical points of the functional. Next, boundary conditions are applied to determine the specific solution that satisfies the optimization problem. Finally, the solution is evaluated and compared to other potential solutions to determine the optimal surface shape.

4. What are some common applications of solving surface optimization problems with calculus of variations?

The calculus of variations is used in a variety of fields, including physics, engineering, and economics. Some common applications include determining the shape of a soap bubble, finding the optimal path for a spacecraft, and minimizing surface area in biomimicry designs.

5. What are the limitations of using calculus of variations to solve surface optimization problems?

The calculus of variations can only be used to solve problems with a single independent variable, such as the shape of a curve or surface. It also assumes that the functional is continuous and differentiable. Additionally, it may not always provide a unique solution, and numerical methods may be required for more complex problems.

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