Finding Extremals of a Functional

[LagrangeEuler]

Homework Statement

Find extremals of the functional
$\Phi(y,z)=\int^{\frac{\pi}{2}}_0((y')^2+(z')^2+2yz)dx$
for
$y(0)=0$, $y(\frac{\pi}{2})=1$, $z(0)=0$, $z(\frac{\pi}{2})=-1$

Homework Equations

The Attempt at a Solution

Well I have a solution but I have problem how to start with it
Solution
System of equation
$y''-z=0$
$z''-y=0$
Differentiating first equation two times and add second equation two given result we obtain
$y^{(4)}-y=0$
$y=C_1e^x+C_2e^{-x}-C_3\cos x+C_4\sin x$
And from boundary condition
$C_1=C_2=C_3=0$
$C_4=1$
We obtain extremals
$y=\sin x$ and $z=-\sin x$
My problem is how to get this system of equations. Tnx for your help.

 

[Dick]

Homework Statement

Find extremals of the functional
$\Phi(y,z)=\int^{\frac{\pi}{2}}_0((y')^2+(z')^2+2yz)dx$
for
$y(0)=0$, $y(\frac{\pi}{2})=1$, $z(0)=0$, $z(\frac{\pi}{2})=-1$

Homework Equations

The Attempt at a Solution

Well I have a solution but I have problem how to start with it
Solution
System of equation
$y''-z=0$
$z''-y=0$
Differentiating first equation two times and add second equation two given result we obtain
$y^{(4)}-y=0$
$y=C_1e^x+C_2e^{-x}-C_3\cos x+C_4\sin x$
And from boundary condition
$C_1=C_2=C_3=0$
$C_4=1$
We obtain extremals
$y=\sin x$ and $z=-\sin x$
My problem is how to get this system of equations. Tnx for your help.

You just use the Euler-Lagrange equation to derive that system, LagrangeEuler. Can you state what that is to get started anyway?

 

[LagrangeEuler]

Tnx. I solved the problem.