Huygens and Bernoulli's brachistochrone - Rijksuniversiteit Groningen

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H&B – p.6/27Commentary1. Potential energy: V (x) = 1 2 ω2 x 2 :harmonic oscillatorx ′′ = −ω 2 x = − dVdx (x)2. Pendulum x ′′ = −ω 2 sinx, ω = √ g/lan-isochronous: period −→ ∞ as x → π3. Gestalt-switch: bead moves along wire profilein vertical plane (gravitation, no friction)pendulum ←→ circleWhat profile provides harmonic oscillations(hence isochrony)?
Parametrization cycloidRoll wheel ( radius ̺) along ceiling −→ξ(ϕ) = ̺(ϕ + sin ϕ),η(ϕ) = ̺(1 − cosϕ)parameter ϕ called rolling angleH&B – p.7/27
Page 1 and 2: H&B - p.1/27Huygens and Bernoulli
Page 3 and 4: H&B - p.3/27Huygens and BernoulliCh
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 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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