- #1

Bafsemann

- 4

- 0

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter Bafsemann
- Start date

- #1

Bafsemann

- 4

- 0

- #2

HallsofIvy

Science Advisor

Homework Helper

- 43,021

- 970

- #3

saltydog

Science Advisor

Homework Helper

- 1,591

- 3

Bafsemann said:

Guys, I'm just interested in this Ok; know very little about the Calculus of variation. Could someone at least formulate the problem? Let me begin with a conceptual approach:

Assume we have the shape of the boat in the form of a surface above the x-y plane. Now it seems to me that this surface can be a very nicely behaved function. The speed of the boat, I'll just guess will be a function of frictional forces which in turn are dependent on the shape of the surface as well as the boundary which I'll call [itex]\Omega[/itex]. Now, let me take a leap of faith:

We wish to minimize the following integral which sums up all the frictional forces on the hull:

[tex]I=\int\int_{\Omega} G[x,y,f(x,y),f_x,f_y]ds[/tex]

Where G is some functional relationship of these forces to the shape of the hull (and I suppose it's slopes as well) and our objective is to find f(x,y) which minimizes the integral (constrained by realistic limits of course, such as manufacturing ones).

I know for the simple case of a function of a single variable f(x), wishing to minimize the integral:

[tex]I=\int_a^b F(x,y,y')dx[/tex]

we can do some calculus and come up with Euler's equation which must be satisfied:

[tex]\frac{\partial F}{\partial y}-\frac{d}{dx} \frac{\partial F}{\partial y'}=0[/tex]

I suppose there is an equivalent one for a double integral? How about for just any old integral of that form say for:

[tex]I=\int_0^1\int_0^1 (f+f_x+f_y)dxdy[/tex]

How would I find the function f(x,y) which minimizes (or maximized) this integral)? Am I getting off-subject?

Share:

- Replies
- 6

- Views
- 258

- Replies
- 2

- Views
- 140

- Last Post

- Replies
- 4

- Views
- 303

- Replies
- 6

- Views
- 460

- Replies
- 6

- Views
- 1K

- Last Post

- Replies
- 3

- Views
- 1K

- Replies
- 8

- Views
- 994

- Replies
- 0

- Views
- 332

- Replies
- 2

- Views
- 638

- Last Post

- Replies
- 1

- Views
- 342