Given two points in space, the 2D Brachistochrone problem could be solved to give solution of a cycloid. I am wondering how could one prove that in arbitrary dimensions ($d\geq 3$) with a 1D uniform gravity, the problem can always be reduced to a 2D problem?
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$\begingroup$ Hint: the acceleration of the object in the direction of gravity is enough to determine travel time. Thus you can construct a curve with equal travel time on a 2D plane $\endgroup$– ioveriCommented Apr 28 at 10:39
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The horizontal $x$-coordinate in the Brachistochrone functional $$T[y]=\int_0^a \!\mathrm{d}x\sqrt{\frac{1+y^{\prime}(x)^2}{2gy(x)}}$$ can be viewed as the arc length parameter of an arbitrary fixed/given smooth horizontal curve $C$ in the horizontal space $\mathbb{R}^{d-1}$ between the given initial and final horizontal points.
In other words, the Brachistochrone problem is intrinsic to the 2D manifold $C\times \mathbb{R}$, where $\mathbb{R}$ is vertical space (with coordinate $y$).