Intuition on a constrained optimization problem

Question

I've solved a) and b), the problem I'm having is with c), I don't intuitively see why there must be an infinite number of solutions to this problem. The only thing I intuitively can see is that I describes arc length for y.

Answer 1

Notice that we can re-write the energy functional as, $$ J(y) = \int_{y(0)}^{y(1)} A(y)dy = \int_0^1 A(z)dz = J $$ which clearly does not depend on $y$ and hence there will be infinite solutions for the minimization problem (if there are any, and we know there are because $A$ is smooth).

In this problem we have to integrate the smooth function $A:\mathbb{R}\rightarrow \mathbb{R}$ over a line which is parametrised by $y:\mathbb{R} \rightarrow \mathbb{R}$, $x\mapsto y(x)$. The fact that the length of that line is $L>\sqrt{2}$ says that this is an admissible line in the Euclidean space $\mathbb{R}^2$ that goes from $(0,0)$ to $(1,1)$, since the minimum length of such a line with an Euclidean metric is of course $L=\sqrt{2}$.