On true brachistochrones or lines of the fastest descent in a resistant medium
Euler again considers friction. He takes a rigorous variational approach to determine the optimal curve, assuming that the friction takes the form hvn+1. He evokes his isoperimetric theorem, but mentions that he has to refine this theorem in order to be able to apply it to the case where velocity depends on arc length – a situation where friction is considered. He succeeds and produces a very generalized version of our modern-day Euler-Lagrange equations, after which he finds that the brachistochrone equation reads (n+2) dv/(vv) − (n+1)C dv √(1+pp)/(pv) + C(1−(h/g)vn+1√(1+pp))∙d(√(1+pp)/p) = 0, where C is some constant, g is half the gravitational acceleration and p=dy/dx. He then proceeds to reduce this equation to √(1+pp) − cp/v + (h/g)vn+1(p/θ−1) = 0, where c=1/C and θ is the tangent of the angle of the end of the curve. Euler then considers two cases of the equation: n=0 (the frictionless brachistochrone solution, i.e., a cycloid) and n=-1 (constant friction with relatively simple equations for x and y).
Original Source Citation
Mémoires de l'académie des sciences de St.-Petersbourg, Volume 8, pp. 29-40.
Opera Omnia Citation
Series 1, Volume 25, pp.326-337.