= Brachistochrone problem
The brachistochrone problem asks for the curve along which a particle released from rest reaches a specified target fastest under a uniform gravitational field, without friction. With downward depth $u$, conservation of energy gives speed $\sqrt{2gu}$ and travel time $\int\sqrt{(1+u'^2)/(2gu)}\,dx$. The <Beltrami identity> makes $u(1+u'^2)$ constant, leading to a <cycloid>. If only the horizontal endpoint coordinate $x_0>0$ is fixed, the <natural boundary conditions for a free endpoint> select the half arch ending at its deepest point, with depth $2x_0/\pi$ and minimum time $\sqrt{\pi x_0/g}$.
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