The remarkable properties of the discrete brachistochrone

Abstract

We present a new solution to the discrete brachistochrone problem, based on the variational principle. There are two interesting and surprising properties of the solution. First, the sliding times along all the straight segments are equal and are independent of the initial velocity. Secondly, we find a simple relation between the angle of the kth segment to the vertical and the angle to the vertical of the first segment. Based on these results, a Matlab code was written that calculates efficiently and accurately the solution for N segments between arbitrary end points.

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

“…This proof bypassed the cumbersome proof based on the Euler-Lagrange equation. By adopting a new approach based upon the results of our previous paper [1], which focused on the discrete brachistochrone, we showed, in two different ways, that the discrete solution converges into the continuous one, i.e. into the cycloid when the number of segments goes to infinity.…”

“…This problem, which initially looks simpler than the original brachistochrone, is actually more difficult to solve analytically, and an analytical solution exists only for the simple case of two segments [9]. In a previous work [1], we showed the general solution of the discrete problem and its remarkable properties listed below:…”

“…where r is the radius of the circle that generates the cycloid (figure 1). The second is a relation between θ k and θ 1 (equation ( 27) in [1]), There are several ways to prove that the discrete path converges in the limit N → ∞ into the continuous path-the cycloid: In this paper, we choose to prove claims (c) and (d).…”

A new solution of the discrete catenary problem

“…In the limit of large N, we showed that the discrete solution converges to the well-known continuous solution. Recently, we used the same strategy in solving the discrete brachistochrone problem and showed that the solution converges to a cycloid, which is the continuous solution [7].…”

“…So, we obtained a very simple but elegant relation between the slopes of the discrete catenary. Once we find q , 1 the remaining angles can be easily calculated from (7). Figure 1 shows that the distance B between the two end points of the chain is equal to the sum of all the projections of the links on the horizontal axis.…”