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.

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