Suppose that $D$ is an open subset of $\mathbb{R}^3$, that $F,G \in \mathcal{C}^2(D)$, and that $L$, $a$, $b$, $c$, $d$ are given real constants with $a<b$
The problem is, to find, amongst all $y \in \mathcal{C}^2([a, b])$ for which
$$ y(a)= c, $$
$$ y(b)= d, $$
$$ (x, y(x),y'(x)) \in D \quad \forall x ∈ [a, b], \quad \text{and} $$
$$ \int_{a}^b G(x, y(x), y'(x)) dx = L, $$
holds, the function $y$, which gives an extreme value (a maximum or a minimum) to the integral
$$ I(y) := \int_{a}^b F(x, y(x), y'(x)) dx.$$
This a the generic case of a variational optimization problem with integral constraints.
On several spots in the literature, I have seen people approach this problem (without further explanation!) using so-called Lagrange multipliers.
I understand the role and mechanism of Lagrange multipliers when optimizing finite-dimensional functions from $\mathbb{R}^n$ to $\mathbb{R}$. However, I do not understand how one can generalize this technique to solve the above problem.
Can someone illustrate the use of Lagrange multipliers for the solution of the above integral-constraint problem and give rigorous mathematical arguments for why it works?