Calculus of variation problem with integral product constraint

Question

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?

Answer 1

1. 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.

2. 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 $$0~=~\frac{\delta\widetilde{J}[y,\lambda]}{\delta y(x)}~=~\frac{\delta J[y]}{\delta y(x)}+\lambda(\frac{\delta F[y]}{\delta y(x)}G[y] + F[y]\frac{\delta G[y]}{\delta y(x)}).\tag{6} $$

3. 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 $\lambda, F^{\ast}, G^{\ast}$.

4. 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).