Al Lehnen Mathematics Department Madison Area Technical College 3550 Anderson Street Madison, WI 53704 (608) 246-6567 alehnen@matcmadison.edu http://my.execpc.com/~aplehnen/al.htm

Abstract:
The branch of applied mathematics known as the "Calculus of Variations" has its origins in the brachistochrone problem posed in 1696 by Johann Bernoulli. The problem of the path taken by a "frictionless" bead accelerated from rest by gravity that would minimize the time of transit is reviewed. A discussion of the existence and uniqueness of the cycloid solution shows that if the ratio of the horizontal distance traveled to the vertical distance traveled exceeds  then the least time path will display an absolute minimum below the terminal point of the trajectory. A complete asymptotic characterization of the solution as a function of this ratio is given. Somewhat surprisingly, the key "technical fact" required is the rather simple expansion of arccos(1-x) for small x. Finally, several "trial" function trajectories are considered and their ability to approximate the least-time cycloid is presented.

The text of the presentation is available at http://matcmadison.edu/alehnen/brach/brach.pdf
A parallel html version is at http://matcmadison.edu/alehnen/brach/brachchistochrone.html

The following Winplot files illustrate various topics from the presentation. Winplot itself can be downloaded at  http://math.exeter.edu/rparris/winplot.html. An online tutorial is also available. To download a file, left click with the mouse and save the file to your local disk. Then you can open and view and/or manipulate the file in Winplot.

Illustration of fitting the initial condition of the cycloid at X = r by varying the parameter k: Two different solutions are shown. Animate on k, m, or r.

A plot of the parameter k versus r = a/L . This illustrates using the asymptotic and Taylor series to explicitly construct the Brachistochrone Solution.

A plot of the parameter Theta Max versus r = a/L . This also illustrates using the asymptotic and Taylor series to construct the Brachistochrone Solution.

A dynamic plot of how the minimizing cycloid varies with  r = a/L . Animate on the parameter r and watch the cycloid change shape.

An illustration of the different linear segment variational solutions tested. Animate on the parameters a, b, d or h to make the path change.

A dynamic plot of how both the minimizing cycloid and the minimized "L" trajectory vary with  r = a/L . Animate on the parameter r.

A dynamic plot of how both the minimizing cycloid and the minimized "Left Ramp" trajectory vary with  r = a/L . Animate on the parameter r.

A plot of how the parameter m (the squared term's coefficient) varies with r = a/L for the least time parabolic trajectory.

A dynamic plot of how both the minimizing cycloid and the minimized parabolic trajectory vary with  r = a/L . Animate on the parameter r.

A dynamic plot of how the minimizing cycloid and all of the the minimized variational trajectories vary with  r = a/L . Animate on the parameter r.

A plot of the time of travel versus r = a/L for the minimizing cycloid and all of the the minimized variational trajectories.

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