Calculus of variations

1. Aug 11, 2005

Callisto

Hi all,

I seeking some advice about the calculus of variations.
I am an undergraduate and i am enrolled in a topic of the above mentioned. After successfully completing the requirments for the topic, 3 weeks after commencement i am feeling way out of my depth. I understand that the calculus of variations is dealing with the optimization of functionals but already the homework assignments have me stumped.
ie: with r as the independent variable, find the polar equation of a straight line by minimizing the integral between the points in a plane P & Q

int:sqrt(dr^2+r^2*dtheta^2)

Geez! where do start?. I have aquired some books on the topic by Bliss, Weinstock and Sagan which at this stage are not much help (no fault of the authors). Can anybody point me in the right direction or offer some advice as to how i should tackle this topic, I AM DETERMINED! to succeed.

Callisto

2. Aug 11, 2005

mathmike

this is a double intergal in which you need to make a worthy substitution in which will replace dtheta

3. Aug 12, 2005

MalleusScientiarum

Hint:
$$\sqrt{dr^2 + r^2 d\theta ^2} = d\theta \sqrt{\left (\frac{dr}{d\theta} \right)^2 + r^2}$$

4. Aug 12, 2005

Callisto

I see that we get

L = int:dtheta*sqrt((dr/dtheta)^2+r^2)

which is the length of the polar curve r=f(theta)

how do i decide what is a worthy substitution for dtheta?

5. Aug 12, 2005

HallsofIvy

Staff Emeritus
Malleus Scientiarum gave you what you needed:

The integral that you want to minimize is $$\int \sqrt{\left(\frac{dr}{d\theta}\right)^2+ r^2}d\theta$$ and the problem is to find r as a function of θ to minimize that integral.

Surely after 3 weeks in the class you know the "Euler-Lagrange" equation?

Last edited: Aug 12, 2005