I have a question.
I want to minimise the functional $$L = \int_{\mathbb{R}}y(t)^2dt.$$ And I have the additional constraints, that $y(t) = y(-t)$ for all $t \in\mathbb{R}$ $$\int_\mathbb{R}y(t)dt = 1,\; \int_\mathbb{R}t^2y(t)dt = 1.$$
The solution is supposed to give the function $y(t) + c_1 + c_2t^2=0$, where $c_1,c_2$ are constants given by the constraints. However, when I try to use the Euler-Lagrange equation for calculus of variations, I get, with $F(t,y,y')=y^2$: $$\frac{\partial F}{\partial y} = 2y, \; \frac{d}{dt}\left(\frac{\partial F}{\partial y'}\right) = \frac{d}{dt} (0) = 0.$$ Which leaves me with $2y = 0$ and thus $y=0$.
I don't know how to use the constraints to get the constants $c_1,c_2$ because I only find $y=0$ to be a solution.
If anybody could point out what I am doing wrong, I would be very grateful.