English Title
On the brachistochrone in a resistant medium while a body is attracted to a center of forces in one way or another
Enestrom Number
761
Fuss Index
382
Original Language
Latin
Content Summary
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.
Topics
Calculus of Variations
Published as
Journal article
Published Date
1822
Written Date
1780
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.
Record Created
2018-09-25
