Tautochrone Problem / Brachistochrone

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The tautochrone problem addresses finding a curve down which a mass placed anywhere on the curve will reach the bottom in the same amount time, assuming uniform gravity. The solution, discovered in May 1697 by at least five different mathematicians, is an (inverted) cycloid [1]. A similar problem is the brachistochrone problem, which asks the question: What is the curve of fastest descent? The solution is also a cycloid.

An inverted cycloid is the shortest path which a ball will roll down in the shortest time [2].

History of the Tautochrone Problem

The tautochrone problem goes back to the time of Galileo (1564-1642), who discovered that large, high-speed pendulum swings or small, low-speed swings take about the same length of time. He later realized he could use the principle to construct a clock, but he lacked the mechanical skills needed to actually build one. Later on, Christiaan Huygens (1629-1695) constructed the first working model of the cycloidal pendulum. Several of his pendulum clocks to determine longitude at sea were built and tested in 1662. In his book, Horologium Oscillatorium, Huygens proved that the cycloid is tautochronous [3], which means “occupies the same time”.

During his study of pendulums, Huygens discovered that a ball rolling back and forth on an inverted cycloid completes a full “swing” in exactly the same amount of time. He solved the tautochrone problem without calculus, using Euclidean geometry [4]. In addition to the cycloid, there are an infinite number of tautochrone curves which are solutions to the problem [5]. One way to solve the problem is with Laplace Transforms [6].

References

[1] Malo, R. (2016). The Tautochrone Problem. Retrieved April 11, 2021 from: https://math.montana.edu/malo/172f16/Tautochrone.pdf
[2] Larson, R. & Edwards, B. (2009). Calculus, 9th Edition. Brooks Cole.
[3] MacTutor. Christiaan Huygens. Retrieved April 11, 2021 from: https://mathshistory.st-andrews.ac.uk/Biographies/Huygens/#:~:text=Huygens%20was%20elected%20to%20the,with%20a%20spring%20regulated%20clock.
[4] Swift, J. Cycloid. Retrieved April 11, 2021 from: https://oak.ucc.nau.edu/jws8/dpgraph/cycloid.html
[5] Pedro, T. et al. Is the tautochrone curve unique? Retrieved April 11, 2021 from: https://ui.adsabs.harvard.edu/abs/2016AmJPh..84..917T/abstract
[6] Gulas, M. (2018). The Pope’s Rhinoceros and Quantum Mechanics. Retrieved April 11, 2021 from: https://scholarworks.bgsu.edu/cgi/viewcontent.cgi?article=1469&context=honorsprojects

CITE THIS AS:
Stephanie Glen. "Tautochrone Problem / Brachistochrone" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/tautochrone-problem-brachistochrone/
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