Euler-Lagrange Equation for Several Dependent Variables

[CMJ96]

Homework Statement

[/B]

Homework Equations

$$f_u- \frac{d}{dx} \left(f_{u'} \right) = 0 $$
$$f_v- \frac{d}{dx} \left(f_{v'} \right) = 0 $$

The Attempt at a Solution

So I calculated the following, if someone could check what I've done it would be greatly appreciated, but I'm not convinced this is right
$$f=(u')^2+2uv+(v')^2$$
$$f_u=2v$$
$$f_v=2u$$
$$f_{u'}=2u'$$
$$f_{v'}=2v'$$
Subbing these into the Euler-Lagrange equations I got
$$v-u''=0$$
$$u-v''=0$$
Then I subbed $v=u''$ into $u-v''=0$ to get $u-u''''=0$.
Using $u=e^{\alpha x}$ I got the following expression (not sure if this bit is right)
$$u(x)=Ae^{-x} +Be^{x} +Ce^{ix} +De^{-ix} $$
I'm not sure if I can use the boundary conditions to find the extremals with this? it doesn't look like it would simplify down to a tidy solution?

 

[Ray Vickson]

Homework Statement

View attachment 225283 [/B]

Homework Equations

$$f_u- \frac{d}{dx} \left(f_{u'} \right) = 0 $$
$$f_v- \frac{d}{dx} \left(f_{v'} \right) = 0 $$

The Attempt at a Solution

So I calculated the following, if someone could check what I've done it would be greatly appreciated, but I'm not convinced this is right
$$f=(u')^2+2uv+(v')^2$$
$$f_u=2v$$
$$f_v=2u$$
$$f_{u'}=2u'$$
$$f_{v'}=2v'$$
Subbing these into the Euler-Lagrange equations I got
$$v-u''=0$$
$$u-v''=0$$
Then I subbed $v=u''$ into $u-v''=0$ to get $u-u''''=0$.
Using $u=e^{\alpha x}$ I got the following expression (not sure if this bit is right)
$$u(x)=Ae^{-x} +Be^{x} +Ce^{ix} +De^{-ix} $$
I'm not sure if I can use the boundary conditions to find the extremals with this? it doesn't look like it would simplify down to a tidy solution?

You can easily check for yourself if this is correct: from you formula for $u$ you can get a formula for $v$ as $v = u''$. Then you can check if your $v$ gives you $u = v''$.

Also: get rid of the question marks ("?"). You should be telling us that you are unsure, rather than asking us if you are unsure.

 

[CMJ96]

Ahhhh yes I see! I didn't think of doing that,when I subbed my U value into the equations it worked. Thanks for the tip, simple but effective :)