Historical Approaches and Modern Methods in Analyzing the Brachistochrone Problem

Document Type : Research Paper

Author

Faculty of Engineering, Ardakan University, P.O. Box 184, Ardakan, Iran.

Abstract

This paper presents the history of competition of mathematicians to uncover the characteristics of the cycloid curve, titled "Helen of Geometry``. It also includes historical notes related to the Brachistochrone problem. The cycloid is introduced in parametric form, and its properties and applications are obtained. Additionally, the calculus of variations and the study of the Brachistochrone problem, along with various solutions, are presented. We show that the optimal solution to the Brachistochrone problem is the cycloid curve. Several procedures in Maple software are introduced to provide a more accurate analysis. These graphical outputs help in understanding the yields. The proposed Maple procedures allow for the comparison of the shortest time problem along paths such as lines, circular arcs, and cycloids. These procedures also aid in drawing connecting curves between points and in calculating optimal travel times numerically. The findings clearly demonstrate that the travel time along the cycloid curve is significantly shorter than along other paths.

Keywords

Main Subjects


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