E761 -- De brachystochrona in medio resistente, dum corpus ad centrum virium utunque attrahitur

(On the brachistochrone in a resistant medium while a body is attracted to a centre of forces in one way or another)

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 $$p = \frac{dy}{dx}$$. Using the general isoperimetric theorem, derived in E760, Euler finds

\begin{aligned} \frac{\omega dv}{v} + \frac{\omega dV}{V} - \frac{Cv\omega tdV}{V} + \frac{V\omega - X}{VV}\left(CvtdV - CVvdt - dV - \frac{Vdv}{v}\right) = 0, \end{aligned}

where $$\omega = \sqrt{1 + ppxx}$$, $$t = \frac{\sqrt{1 + ppxx}}{pxx}$$ and $$C$$ is some constant. Reducing this equation, Euler finds $$-\frac{1}{CVv} + \frac{t}{V} - \int\frac{\omega dt}{X} = \Delta$$ for some $$\Delta$$. Using a relation between $$v$$ and $$p$$, this curve can then be explicitly found.

Publication:
• Originally published in Memoires de l'academie des sciences de St.-Petersbourg 8, 1822, pp. 41-45
• Opera Omnia: Series 1, Volume 25, pp. 338 - 342
Documents Available:
• Original publication: E761
• English draft translation (Carlos Hermans): E761