Bernoulli's lightray solution for the brachistochrone problem from ...

A few high school computationsLemma. (abbreviating v ′ = dv/dy)v ′ = − gv(y)andcosα = Sv ′ (y) dydα ,Proof: Differentiate (2) with respect toy and (1) with respect toα□Thusdydαlemma=1Sv ′ (y)cosαlemma= − v(y)Sgvcosα(1)= − 12S 2 gsin(2α) (3)while:dxdα= tanαdydα(3)= − v(y)Sgsinα(1)= − 12S 2 g (1−cos(2α))Integration then gives the desired result.Cycloid with radius14S 2 gand rolling angle 2α□