#### English Title

On true brachistochrones or lines of the fastest descent in a resistant medium

#### Enestrom Number

760

#### Fuss Index

380

#### Original Language

Latin

#### Content Summary

Euler again considers friction. He takes a rigorous variational approach to determine the optimal curve, assuming that the friction takes the form *hv*^{n+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*)*v*^{n+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*)*v*^{n+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*).

#### 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. 29-40.

#### Opera Omnia Citation

Series 1, Volume 25, pp.326-337.

#### Record Created

2018-09-25