Minimizing Area Under Curve: COV

[opsb]

So, If you've got two points and a given length of curve to 'hang' between them, what shape is the curve which minimises the area underneath it? For a curve which is almost the same length as the distance between the points, this would be a catenary, I think (a la famous hanging chain problem), but for longer curves it would be different. Any ideas?

 

[Petr Mugver]

So you want to minimize the integral

$$I=\int_a^b f(x)dx$$

with the constraints

$$f(a)=A$$

$$f(b)=B$$

$$L=\int_a^b\sqrt{1+f'^2}dx$$

It's an Euler-Lagrange problem. The lagrangian is

$$\mathscr{L}=f+\lambda\sqrt{1+f'^2}$$

so the equations of motion are

$$\frac{d}{dx}\frac{\lambda f'}{\sqrt{1+f'^2}}=1$$

in other words

$$\frac{\lambda f'}{\sqrt{1+f'^2}}=cx+d$$

You have to find c, d and lambda using the constraints above. Then you have to solve for f '. Finally you integrate (this is the hard part) to find f.