Huygens and Bernoulli's brachistochrone - Rijksuniversiteit Groningen

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H&B – p.18/27Johann Bernoulli’s surpriseThe brachistochrone problem (Groningen 1696):Given: Two points P and Q in vertical plane underinfluence of gravityRequest: Wire profile for bead to slide from P to Q inthe shortest timeOptical metaphore:bead follows light ray through optical medium withsuitably chosen speed op propagation v = v(h)
H&B – p.19/27Two principlesP1x2α jn jjα j+1n j+1j +1NhQSpeed of propagation by Conservation of Energy12 v2 (h) − gh = constant (1)Lightray by Fermat Principle of least time:
Page 1 and 2: H&B - p.1/27Huygens and Bernoulli
Page 3 and 4: H&B - p.3/27Huygens and BernoulliCh
Page 6 and 7: H&B - p.6/27Commentary1. Potential
Page 8 and 9: H&B - p.8/27Arclength cycloidarclen
Page 10: H&B - p.10/27Evolute and evolventSo
Page 14 and 15: The master himself . . .H&B - p.14/
Page 16 and 17: H&B - p.16/27Detailed formulation1.
Page 20 and 21: H&B - p.20/27DiscretisationHorizont
Page 22 and 23: H&B - p.22/27OpticsPn 1 = 1/v 1α 1
Page 24 and 25: H&B - p.24/27Deduction cycloidSnell
Page 26 and 27: H&B - p.26/27Commentary1. Challenge

Huygens and Bernoulli's brachistochrone - Rijksuniversiteit Groningen

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