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

Laws, properties- Energy conservationḢ≡ 0, say H(q,p) = E- q = (x,y) and p x = n 2 (y)ẋ,p y = n 2 (y)ẏH(x,y,p x ,p y ) = 12n 2 (y) (p2 x +p 2 y)- H independent of x (cyclic variable): ṗ x ≡ 0conservation of momentum, sayp x = I- translation symmetry, Noether principle- conservation ofI √2E(x,y,ẋ,ẏ) = n(y)ẋ√ẋ2+ẏ 2 = n(y)sinα(ẋ,ẏ)(= S for Snell, this is familiar !)