E759 -- Investigatio accuratior circa brachystochronas

(A more accurate investigation into brachistochrones)

Summary: Euler starts by mentioning how he didn't like his achievements in E042. He therefore attempts to derive the equations from scratch, using the "first principles" of motion, which basically consist of Newton's laws of motion. He first considers the two dimensional brachistochrone problem, finding that \(vdv = 2g(Xdx + Ydy)\). Subsequently, using his "isoperimetric treatment" (which is basically a more refined Euler-Lagrange equation) he finds \(\Theta = \frac{v^2}{2gr}\), where \(r\) is the radius of curvature, \(\Theta\) is the normal force and \(g\) is half the gravitational acceleration.

After this, Euler takes a look at the three dimensional brachistochrone problem, where he again works from Newton's laws of motion, except now he uses three coordinates. He splits the problem in two projections, one in the \(xz\) plane and another in the \(xy\) plane, where he solves the brachistochrone for both, and later merges them together. This gives the set of equations

\(\begin{aligned} \frac{Y(1 + qq) - pX - pqZ}{\sqrt{1 + pp + qq}} + \frac{vv}{2gdx}d\left(\frac{p}{\sqrt{1 + pp + qq}}\right) = 0 \\ \frac{Z(1 + pp) - qX - pqY}{\sqrt{1 + pp + qq}} + \frac{vv}{2gdx}d\left(\frac{q}{\sqrt{1 + pp + qq}}\right) = 0, \end{aligned}\)

where \(X\), \(Y\) and \(Z\) are the respective disturbing forces in the \(x\), \(y\) and \(z\) directions, \(p = \frac{dy}{dx}\) and \(q = \frac{dz}{dx}\). He concludes by mentioning that restricting this to two dimensions again yields the same equation as before:
\(\begin{aligned} \frac{Xp - Y}{\sqrt{1 + pp}} = \frac{vv}{2g}d\left(\frac{p}{\sqrt{1 + pp}}\right). \end{aligned}\)

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