If one wants to extremize the integral
$\int_{x_0}^{x_1} F(x, y_i, y_i')dx$
subject to constraints
$\phi_j = g_j(x,y_i,y_i')=0$,
using the calculus of variations,
then one can generate the Euler-Lagrange equations using the modified function
$\Phi= F+\sum_j \lambda_j(x) g_j(x, y_i, y_i')$.
That all makes sense to me. I can develop the right number of differential equations. My question is this. To solve those differential equations (as an initial value problem), one needs initial conditions on the $\lambda_j$:
$\lambda_j(x=0)=?$ (and possibly derivatives of $\lambda_j$ depending on resulting order of the differential equations).
Are there any guidelines for setting these boundary conditions?