# Stationary Point of a functional

• Jan 11th 2009, 08:38 AM
TheEmptySet
Stationary Point of a functional
Okay, I have worked this problem a few times and I end up with a solution that I don't think is correct. Any feedback would be awsome.

If possible, find the stationary point of the functional

$\displaystyle \mathcal{J}(y)=\int_{0}^{1}(y^2+x^2y')dx, y(0)=0, y(1)=\alpha$

So I use the Euler-Lagrange Equation

$\displaystyle \mathcal{L}_y-\frac{d}{dx}\mathcal{L}_{y'}=0$ with

$\displaystyle \mathcal{L}(y,y',x)=y^2+x^2y'$

so $\displaystyle \frac{\partial}{\partial y}(y^2+x^2y')=2y$

and $\displaystyle \frac{\partial}{\partial y'}(y^2+x^2y')=x^2$

and $\displaystyle \frac{d}{dx}\bigg[\frac{\partial}{\partial y'}(y^2+x^2y') \bigg]=\frac{d}{dx}x^2=2x$

So now I end up with

$\displaystyle 2y-2x=0 \implies y=x$

I don't think this is correct and it would only be consistant with the boundry values if $\displaystyle \alpha=1$

Thanks
• Jan 11th 2009, 08:49 AM
Opalg
That looks correct to me. Why do you think it is wrong? After all, the question says "If possible, find the stationary point", and your answer is that it is not possible, unless α=1.
• Jan 11th 2009, 08:56 AM
Jester
Quote:

Originally Posted by TheEmptySet
Okay, I have worked this problem a few times and I end up with a solution that I don't think is correct. Any feedback would be awsome.

If possible, find the stationary point of the functional

$\displaystyle \mathcal{J}(y)=\int_{0}^{1}(y^2+x^2y')dx, y(0)=0, y(1)=\alpha$

So I use the Euler-Lagrange Equation

$\displaystyle \mathcal{L}_y-\frac{d}{dx}\mathcal{L}_{y'}=0$ with

$\displaystyle \mathcal{L}(y,y',x)=y^2+x^2y'$

so $\displaystyle \frac{\partial}{\partial y}(y^2+x^2y')=2y$

and $\displaystyle \frac{\partial}{\partial y'}(y^2+x^2y')=x^2$

and $\displaystyle \frac{d}{dx}\bigg[\frac{\partial}{\partial y'}(y^2+x^2y') \bigg]=\frac{d}{dx}x^2=2x$

So now I end up with

$\displaystyle 2y-2x=0 \implies y=x$

I don't think this is correct and it would only be consistant with the boundry values if $\displaystyle \alpha=1$

Thanks

Usually one gets differential equations (is that what you were expecting?). Check and make sure the Lagrangian is correct.
• Jan 11th 2009, 09:01 AM
TheEmptySet
I was expecting a differential equatin of some kind. Thank again for the feedback.