Derivation of Euler-Lagrange equation?

[mizzcriss]

Homework Statement

Problem 1:

Derive the Euler-Lagrange equation for the function ##z=z(x,y)## that minimizes the functional

$$J(z)=\int \int _\Omega F(x,y,z,z_x,z_y)dxdy$$

Problem 2:

Derive the Euler-Lagrange equation for the function ##y=y(x)## that minimizes the functional

$$J(y)=\int_{a}^{b}F(x,y,y',y'')dx$$

Homework Equations

I know the Euler-Lagrange equation for the functional ##J=\int_{a}^{b}F(x,y,y')dx## is ##\frac{\partial f}{\partial y}-\frac{d}{dx}(\frac{\partial f}{\partial y'})##

The Attempt at a Solution

I've found many resources for the derivation of the Euler-Lagrange for ##J=\int_{a}^{b}F(x,y,y')dx## but I don't know how to apply them in order to derive the Euler-Lagrange for my two problems. When I tried to derive it for Problem 2, I couldn't figure out what to do with the ##y''## term.

I'm not asking anyone to give me the complete derivations because I know that would be insanely time consuming to type out but I don't even know where to begin! This is for a Numerical Methods of ODE's class by the way. I have a final on Tuesday and I know something like this will be on it!

 

[dextercioby]

Derive, derive, derive. Do you have a textbook ? The multimensional case is normally covered there, since it's really different than the 1D one. For the second, you should use the easiest proof for one differential (and one variable) and adjust to take into account the appearance of the 2nd derivative.

Weinstock's 1974 Dover published < Calculus of Variations with Applications to Physics & Engineering > is a reccomendable text.

 

[mizzcriss]

To start, thanks for your help! :-)

The textbook used for the class doesn't cover calculus of variation, my professor covered it separately and only derived the simple case. I have tried to apply the simple case directly to my second problem but I don't know what to do with the ##y'## and ##y''## integrals once I've integrated them by parts. I would totally have ordered that book if I had like another week before my final but there's no way I could get it in time for my final :-D

Do you know of any sites that would have anything similar to what I'm trying to do?