How can I find the critical curves for the following functional

Question

Find the critical curves for the following functional : $$J[y,z]=\int_{0}^{1} \sqrt{1+y'^2+z'^2}$$ such that :$$y^2+z^2=1$$ and $$y(0)=z(1)=1$$ $$y(1)=z(0)=0$$

Answer 1

Write $y(t)=\cos\big(x(t)\big)$ and $z(t)=\sin\big(x(t)\big)$. Therefore, we may take $x(0)=0$ and $x(1)=\left(2n+\frac12\right)\pi$, for some integer $n$. The integral becomes $\int_0^1\,\sqrt{1+\big(x'(t)\big)^2}\,\text{d}t$, which calculates the length of the curve $\big(t,x(t)\big)$ from $t=0$ to $t=1$. The critical curves are therefore straight lines: $x(t)=w_nt$, where $w_n:=\left(2n+\frac12\right)\pi$ for each integer $n$. Hence, for the original functional, the critical curves are given by $$\big(y(t),z(t)\big)=\Big(\cos\left(w_nt\right),\sin\left(w_nt\right)\Big)$$ for all integers $n$.