On the brachistochrone in a resistant medium while a body is attracted to a center of forces in one way or another
Euler takes a look at the friction brachistochrone, where the force attracts a body to some point in space. Using polar coordinates, Euler calls O the attraction point and X an arbitrary point for which the centripetal force is x. Lastly he defines y, the angle between the initial point A, O and X, with p=dy/dx. Using the general isoperimetric theorem, derived in E760, Euler finds that ω dv/v + ω dV/V − Cvωt dV/V + [(Vω−X)/(VV)]∙(Cvt dV − CVv dt − dV − V dv/v) = 0, where ω=√(1+ppxx), t=√((1+ppxx)/(pxx)) and C is some constant. Reducing this equation, Euler finds -1/(CVv) + t/V − ∫ω dt/X = Δ for some Δ. Using a relation between v and p, this curve can then be found explicitly.
Original Source Citation
Mémoires de l'académie des sciences de St.-Petersbourg, Volume 8, pp. 41-45.
Opera Omnia Citation
Series 1, Volume 25, pp.338-342.