E760 -- De vera brachystochrona seu linea celerrimi descensus in medio resistente

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

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, whence he finds that the brachistochrone equation reads

\(\begin{aligned} \frac{(n + 2)dv}{vv} - \frac{(n + 1)Cdv\sqrt{1 + pp}}{pv} + C\left(1 - \frac{h}{g}v^{n + 1}\sqrt{ 1 + pp}\right)d\left(\frac{\sqrt{1 + pp}}{p}\right) = 0, \end{aligned}\)

where \(C\) is some constant, \(g\) is half the gravitational acceleration and \(p = \frac{dy}{dx}\). He then proceeds to reduce this equation to \(\sqrt{ 1 + pp} - \frac{cp}{v} + \frac{h}{g}v^{n + 1}\left(\frac{p}{\theta} - 1\right) = 0\), where \(c = \frac{1}{C}\) and \(\theta\) is the tangent of the angle of the end of the curve. Euler then considers two cases of the equation, the first of which is \(h=0\), which gives the frictionless brachistochrone solution: a cycloid. The latter is \(n=-1\), so that friction is constant and thus the equations for \(x\) and \(y\) become relatively simple. He ends by concluding how his equations generalize for any desired form of friction.

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