Volume 18, Issue 3 (Journal of Control, V.18, N.3 Fall 2024)
JoC 2024, 18(3): 49-58 |
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Kafash B. An Innovative Solution for the Brachistochrone Problem as a Class of the Calculus of Variations. JoC 2024; 18 (3) :49-58
URL: http://joc.kntu.ac.ir/article-1-1031-en.html
URL: http://joc.kntu.ac.ir/article-1-1031-en.html
Ardakan University
Abstract: (927 Views)
The problem of the shortest time is an old problem that is referred to as the origin of the calculus of variation. The various applications of calculus of variations in different fields, including mathematics, physics, electrical engineering, mechanics, and robotics, have transformed this area into an active research field for mathematicians and engineers. In this article, an innovative method for solving the Brachistochrone Problem is proposed. This method is based on transforming the differential equation of the Brachistochrone Problem into a system of differential equations. In the proposed approach, two non-trivial choices for forming the system of equations are presented, and the general form of the solution is obtained in each case. Finally, it is determined that only one of the two obtained solutions satisfies the proposed final condition of the problem. Additionally, the optimal solution obtained from the proposed path exactly matches the value obtained along the optimal path, namely the cycloid. Furthermore, two Maple procedures are presented, and the resulting outputs are displayed.
Keywords: The Brachistochrone Problem, Calculus of variations, cycloid curve, Euler-Lagrange differential equation
Type of Article: Research paper |
Subject:
Special
Received: 2024/07/2 | Accepted: 2024/11/7 | ePublished ahead of print: 2024/11/22 | Published: 2024/12/20
Received: 2024/07/2 | Accepted: 2024/11/7 | ePublished ahead of print: 2024/11/22 | Published: 2024/12/20
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