Math Fest 2008

The Brachistochrone Revisited: A Timely Consideration

Thursday, July 31, 2008

Al Lehnen
Mathematics Department
Madison Area Technical College
3550 Anderson Street
Madison, WI 53704


In June 1696 Johann Bernoulli published as a challenge the following problem. "Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time." This challenge, the brachistochrone, is one the origins of the Calculus of Variations. The problem is to determine for a frictionless bead accelerated from rest the path that would minimize the time of descent between two specified points. The well-known solution is an inverted cycloid. The fundamental independent variable of the problem is r, the ratio of the horizontal to vertical displacement between the two points. Properties of the cycloid solution as r varies, including its uniqueness, are elaborated. In particular, if r exceeds  , then the least time path attains an absolute minimum below the terminal point of the trajectory.

Figure 1: Animation of the Minimizing Cycloid versus xr

In this presentation explicit asymptotic expansions for the time of descent for both large and small r are developed. Relatively simple and computationally efficient formulas that allow for an accurate graph of the minimizing cycloid for any value of r are derived. In addition, variational calculations that minimize the time of descent for trial function trajectories are presented. The first trajectories considered were piecewise linear segments. Their ability to approximate the solution of the brachistochrone is summarized. Finally, a rather thorough analysis of a minimizing parabola is given. Despite the fact that having  y descend as a quadratic function of x between the two points leaves only one free parameter, such a trajectory is able to come very close in matching the time of descent of the minimizing cycloid.

Figure 2: Animation of the Minimizing Cycloid and the Minimizing Parabola versus xr

Figure 3: Animation of the Minimizing Cycloid and the Variational Solutions versus xr

A comparison of the total time of travel between the different variational methods and the minimizing cycloid is displayed in Figure 4. The notation is that T(1, r, 0, r) is the straight line trajectory, T(gamma1(r), 0, 0, r) is the “L” trajectory and T(gamma2(r), 0, 0, r) is the “Left Ramp” trajectory. For the least-time parabolic trajectory the discrete points represent the approximate solutions obtained by Newton’s method.

Figure 4

    A summary with results is available at .

    A more detailed version with derivations is .