Huygens and Bernoulli's brachistochrone - Rijksuniversiteit Groningen

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H&B – p.22/27OpticsPn 1 = 1/v 1α 1RδR ′α 2n 2 = 1/v 2QRefraction between two media:n j = 1/v j ,j = 1, 2 refraction indicesSnell’s Law:n 1 sinα 1 = n 2 sinα 2
H&B – p.23/27Optics ctd.δsinα 1δsinα 2n 1 = 1/v 1Rα 1δα 2R ′n 2 = 1/v 2Proof: Take variation of an arbitrary path P → R → Q to P → R ′ → Q with R ′ − R = δTime difference:δn 1 sin α 1 − δn 2 sin α 2 + O(δ 2 )Optimal choice for R :n 1 sin α 1 = n 2 sin α 2QEDComment: A high school proof exists with rectangles . . .
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 18 and 19: H&B - p.18/27Johann Bernoulli’s s
Page 20 and 21: H&B - p.20/27DiscretisationHorizont
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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