How can the Brachistochrone problem be solved if we set the slope at both end points to be 0?

Question

If the the two end points are $(0,0)$ and $(x_1, y_1)$, then $y'(0)=0$ and $y'(x_1)=0$. I have no idea how this constraint can be set up while solving the Euler-Lagrange Equation. Also, is there some general method for optimizing functionals with similar single point constraints?

Answer 1

Recall the fact that solution curves to the Brachistochrone problem are strictly convex functions. (Heuristically, we can save a bit of time by bending a straight line infinitesimally downwards. We leave it to the reader to give a mathematically precise proof.)

This means that the slope is a strongly increasing function. Therefore demanding a zero slope in both endpoints are two mutually incompatible boundary conditions.

Answer 2

The brachistochrone problem is a workhorse in calculus of variations. (You can look it up in many textbooks.) It is usually stated as $$\min \int_{x_0}^{x_1} \left(\frac{1 + (y^\prime)^2}{y} \right)^{1/2} dx$$ Since the integrand $F(x, y, y^\prime)$ does not explicitly depend on $x$, the Euler equation reduces to the special case $$F-y^\prime F_{y^\prime} = \mbox{constant}$$ and the resulting equation can be solved by separating variables.