Huygens and Bernoulli's brachistochrone - Rijksuniversiteit Groningen

H&B – p.25/27A few more high school computationsLemma (abbreviating v ′ = dv/dh)v ′ = g vandcos α = Cv ′ dhdα ,Proof: Differentiate (1) with respect to h and (2) with respect to αQEDThuswhile:dxdαIntegration then givesdhdαlemma== tan αdhdα1Cgv cos α(2)= 12C 2 g(3)= v sin α (2)= 12C 2 (1 − cos2α)gsin2α (3)h(α) = h 0 − 14C 2 g cos2αx(α) = x 0 + 14C 2 (2α − sin2α) :gcycloid with rolling angle ϕ = 2α and radius ̺ = 1/(4C 2 g)QED