I'm currently working on Calculus of Variations and I came across an integral which I had to minimize. The integral I have to minimize is $$\int_0^1(1+y'^2)^2\,dx$$ After getting the Euler-Lagrange equation for this it has 3 solutions. One real and two imaginary solutions. Now my question is that are the complex solutions meaningful solutions since we're looking for a function that minimizes the integral, which is real.
Physically it would be meaningful to choose only the real solution but this is purely a mathematics course, so are complex solutions to EL-equation regarded as valid solutions?
Edit: The EL-equation I got is: $$4y''(1+3y'^2)=0$$ Which gives me $$y(x)=c_1x+c_2,\quad \text{or} \quad y(x)=\pm \frac{i}{\sqrt3}x+c_3$$
First of all, note how your second solution is merely a special case $c_1=\pm\frac i{\sqrt3}, c_2=c_3$ of the first solution. So you might as well pick $c_1=i$ to obtain a vanishing integral, which would be quite minimal. But you should stick with real values and thus plug your first solution into the integral and minimize it with respect to $c_1$ (spoiler: $c_1=0$).
However, once you do allow for complex values, $\int_0^1(1+y'^2)^2\,dx$ no longer makes sense for minimization, since that integral can become complex-valued and thus there is no order by which to determine what is "minimal". You need to involve absolute values, e.g. either $\int_0^1(1+|y'|^2)^2\,dx$ or $\int_0^1|1+y'^2|^2\,dx$. And since you're dealing with complex functions now, $y$'s complex conjugate $\bar y$ is an independent function (alternatively you can split $y$ into real and imaginary part to obtain to independent real valued functions), i.e. you also have to consider $-\frac d{dx}\frac{\partial}{\partial \bar y'}$ to obtain a second Euler-Lagrange equation, coupled to the other one.