The problem to solve is the minimization of a functional of two functions, $F(y,z) = \int_a^b f(y,z)dx$ , subject to a constraint $g(y,z,y',z') = 0$. The augmented functional is then $L(y,z,y',z') = \int_a^b [f + \lambda g]dx$. $\lambda$, y and z are functions of x.
The Euler Lagrange equations lead to the coupled differential equations:
$ g(y,y') = 0 $
$ \frac{\partial f}{\partial z} + \lambda \frac{\partial g}{\partial z} - \lambda \frac{\partial}{\partial x} \frac{\partial g}{\partial z'} - \frac{\partial g}{\partial z'} \lambda '=0$
$ \frac{\partial f}{\partial y} + \lambda \frac{\partial g}{\partial y} - \lambda \frac{\partial}{\partial x} \frac{\partial g}{\partial y'} - \frac{\partial g}{\partial y'} \lambda '=0$
I have, as usual in functional minimization problems, two boundary conditions for y and z. So I guess I have two options:
1 - solving the three equations in the three unknown functions y, z and $\lambda$. But in this case how do I get boundary conditions for $\lambda$ from the ones I have?
2- From the last two equations I can get rid of $\lambda$, by solving for $\lambda$ and $\lambda$' and equating the derivative of $\lambda$ to $\lambda$'. This results in a second order differential equation for y and z to be solved simoultaneously with the constraint. The problem here is that this differential equation is very complex, it involves quotients of polynomials in y, z, y' and z' which seems to be quite untractable...
So what would be the best method to solve this problem?