From the discrete to the continuous brachistochrone: a tale of two proofs

In a previous paper (2019 Eur. J. Phys. 40 035005) we showed how to design a discrete brachistochrone with an arbitrary number of segments. We have proved, numerically and graphically, that in the limit of a large number of segments, N >> 1, the discrete brachistochrone converges into the continuous brachistochrone, i.e. into a cycloid. Here we show this convergence analytically, in two different ways, based upon the results we obtained from investigating the characteristics of the discrete brachistochrone. We prove that at any arbitrary point, the sliding bead has the same velocity on both the continuous and discrete paths, and the radius of the curvature of both paths is the same at corresponding points. The proofs are based on the well-known fact that the curve of a cycloid is generated by a point attached to the circumference of a rolling wheel. We also show that the total acceleration magnitude of the bead along the cycloid is constant and equal to g, whereas the acceleration vector is directed toward the center of the wheel, and it rotates with a constant angular velocity.

Automated summary

This paper demonstrates the convergence of a discrete brachistochrone into a continuous brachistochrone, known as a cycloid, as the number of segments approaches infinity. The proofs are based on the characteristics of the discrete brachistochrone, showing that at any point, the sliding bead has the same velocity and curvature radius on both paths. The acceleration magnitude along the cycloid is constant and equal to g, with the acceleration vector directed towards the center of the wheel.