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Bernoulli’s light ray solution of the brachistochrone problem through Hamilton’s eyes. (English) Zbl 1300.01014

Summary: This article contains a review of the brachistochrone problem as initiated by Johann Bernoulli in 1696–1697. As is generally known, the cycloid forms the solutions to this problem. We follow Bernoulli’s optical solution based on the Fermat principle of least time and later rephrase this in terms of Hamilton’s 1828 paper. Deliberately an anachronistic style is maintained throughout. Hamilton’s solution recovers the cycloid in a way that is reminiscent of how Newton’s mathematical principles imply Kepler’s laws.

MSC:

01A45 History of mathematics in the 17th century
78A05 Geometric optics
49-03 History of calculus of variations and optimal control
49K15 Optimality conditions for problems involving ordinary differential equations
47J30 Variational methods involving nonlinear operators

Biographic References:

Bernoulli, Johann; Hamilton, William Rowan
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References:

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