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- W2325295098 abstract "The brachistochrone problem asks for the curve between two points down which an object will slide in the least amount of time under the force of gravity and neglecting friction. Johann Bernoulli challenged his peers with this problem in June of 1696, and received solutions from Jacob Bernoulli (his brother), Gottfried Leibniz, and Isaac Newton [1]. The mathematical techniques used to solve this and related problems have varied greatly over the last 300 years, from the original geometric arguments to the calculus of variations to graph theory [1, 2, 3]. In this article we solve the problem using concepts and tools from the first years of undergraduate mathematics. Our approach is analogous to finding minima and maxima of ordinary functions. First, we derive an integral which gives the amount of time it would take for an object to slide down any curve under the force of gravity but without friction. Next, we expand this integral in a manner similar to a Taylor series. Then, we recognize that the minimum can be found by ensuring that small deviations only increase the value of the integral. This results in a differential equation whose solution is a cycloid. The brachistochrone problem receives credit for prompting the development of the calculus of variations. Just as ordinary calculus can be used to minimize and maximize functions, the calculus of variations can be used to minimize and maximize functions of functions (so-called functionals). Our solution to the brachistochrone is similar to using the definition of the derivative limh0o[f(x + h) f(x)]/ h. Just as this definition gets replaced by shortcuts, for example d/dx xn = nxn-1, our solution anticipates the Euler-Lagrange formula." @default.
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- W2325295098 date "2004-05-01" @default.
- W2325295098 modified "2023-09-23" @default.
- W2325295098 title "The Brachistochrone Problem" @default.
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- W2325295098 doi "https://doi.org/10.1080/07468342.2004.11922072" @default.
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