Calculus of variation problem with integral product constraint I want to find the extrema of the functional,
$$
J[y] =\int^1_0 L(x, y(x)) \ \text{d} x
$$
in the space of continuous functions $C[0, 1]$, subject to the constraint,
$$
 \int^1_0 \left(y(x) - \frac{1}{2}\right)^2 \text{d} x \ \times 
 \int^1_0 y(t) \ \text{d} t  = 1
$$
I know how to derive the Euler-Lagrange equation for the case in which the constraint is a simple integral of the form
$$
\int^1_0 M(t, y(t), y^\prime(t)) \ \text{d} t = c
$$
but I could not find any reference on a constraint with a product of integrals, as is my case.
Do you know of any reference book I could consult on this or could you point me in the right directions to derive the EL equation myself?
 A: *

*OP's variational problem is of the form
$$ J[y]~=~\int_0^1 \! \mathrm{d}x~L(x,y(x),\ldots) \tag{1}$$
with the constraint
$$F[y]G[y]~=~1,\tag{2}$$
where
$$ F[y]~=~\int_0^1 \! \mathrm{d}x~f(x,y(x),\ldots) \tag{3}$$
and
$$ G[y]~=~\int_0^1 \! \mathrm{d}x~g(x,y(x),\ldots) \tag{4}$$
are functionals.


*Introduce Lagrange multiplier $\lambda$ and an extended functional
$$\widetilde{J}[y,\lambda]~=~J[y]+\lambda(F[y]G[y]-1).\tag{5} $$
Provided adequate boundary conditions, the Euler-Lagrange equation takes the form
$$0~=~\frac{\delta\widetilde{J}[y,\lambda]}{\delta y(x)}~=~\frac{\delta J[y]}{\delta y(x)}+\lambda(\frac{\delta F[y]}{\delta y(x)}G[y] + F[y]\frac{\delta G[y]}{\delta y(x)}).\tag{6} $$


*To solve eq. (6) replace $F[y],G[y]$ with 2 parameters $F^{\ast}, G^{\ast}\in \mathbb{R}$ as follows:
$$0~=~\frac{\delta J[y]}{\delta y(x)}+\lambda(\frac{\delta F[y]}{\delta y(x)}G^{\ast} + F^{\ast}\frac{\delta G[y]}{\delta y(x)}).\tag{7} $$
Eq. (7) is an equation for $y$ as a function of 3 external parameters $\lambda, F^{\ast}, G^{\ast}$.


*Finally, determine possible $\lambda,F^{\ast}, G^{\ast}$ by the self-consistency conditions
$$ F[y]~=~F^{\ast}, \qquad  G[y]~=~G^{\ast}, \tag{9} $$
and the constraint (2).
