Calculus of variations with isoparametric constraint

[MisterX]

We seek stationary solutions to
$\int_{x_0}^{x_1} F(x, y, y')dx$
subject to the constraint
$\int_{x_0}^{x_1} G(x, y, y')dx = c$
where $c$ is some constant.

I have read that this can be solved by applying the Euler Lagrange equations to
$F(x, y, y') + \lambda G(x, y, y')$
and then finding the appropriate value of $\lambda$ when solving so that the constraint is satisfied.

Why does this work? I am not sure what reference to use.

Also, this may still work when the unconstrained integral has no stationary solutions, right?

 

[Greg Bernhardt]

I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?