Huygens and Bernoulli's brachistochrone - Rijksuniversiteit Groningen

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H&B – p.8/27Arclength cycloidarclength x = x(ϕ) (using Pythagoras):dx = √ dξ 2 + dη 2 == √ (dξ/dϕ) 2 + (dη/dϕ) 2 dϕ == ̺√2 √ 1 + cos ϕ dϕ = 2̺ cos ϕ 2 dϕ x(ϕ) = 4̺ sin ϕ 2
Cycloidal wire profileVertical heightη(ϕ) = 2̺ sin 2 ϕ 2 = 18̺(x(ϕ))2 potential energy “V = mgη” : V (x) = mg8̺ x2−→ equation of motion beadx ′′ = − g 4̺x :a harmonic oscillator with ω = √ g/(4̺)Conclusion: cycloid isochronous curve(also tautochronous ...)H&B – p.9/27
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 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 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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