Understanding the derivation of the brachistochrone equation I'm studying theoretical mechanics from "Classical Mechanics" by John Taylor, and I'm a bit stuck on one of the examples. In this example, Taylor goes through multiple steps a bit faster than I'd prefer. Namely:
He takes the equation $\frac{x'^2}{y(1+x'^2)}=C$ and arbitrarily determines the constant $C$ to be equal to $\frac{1}{2a}$ for some $a$ without explanation. We then get $x'=\sqrt{\frac{y}{2a-y}}$, he then states that we should do a substitution $y=a(1-\cos\theta )$, and says that we will get $x=a\int(1-\cos\theta)\,d\theta$.
I have spent multiple hours trying to justify the former step and show the latter step, I'm not sure what I have missed. Is there some trigonometric substitution I'm forgetting?
 A: If we choose two points $P_1=(0,0)$ and  $P_2=(x_2,y_2)$, with the $y$−axis pointing in the same direction as the acceleration due to gravity, $g$. Then we seek to minimize
$$T=\int_{P_1}^{P_2} \frac{ds}{v}$$
where the arc length element along the curve we seek is $ds=\sqrt{dx^2+dy^2}$ and the speed along the curve, $v$ satisfies $\frac 12 mv^2=mgy$ by conservation of energy. Then the functional to be minimized is $$T = \int_{P_1}^{P_2}\sqrt{\frac{1+y'^2}{2gy}}\,\mathrm{d}x$$
where $y'=dy/dx$. The function to be varied in this minimization,
$$F(x,y,y') = \sqrt{\frac{1+y'^2}{2gy}}$$
does not explicitly depend on $x$ so the Beltrami identity applies:
$$F - y'\frac{\partial F}{\partial y'} = C = \mathrm{constant}.$$
Now,
$$\frac{\partial F}{\partial y'} = \frac{y'}{\sqrt{2gy}\sqrt{1+y'^2}}$$
and substitution into the Beltrami identity gives
$$y(1+y'^2) = \frac{1}{2gC^2} = 2a. \tag 1$$
I think that the value of $C$ could be appropriately taken from the $(1)$ or you can consider a parabola with the same points $P_1$ and $P_2$ with vertical tangent at $P_1$ has the formula $y = \sqrt{Cx}$ with $C = (y_2/x_2)^2$. The integral giving $T$ may be evaluated numerically using this and $y'=C/(2y)$ with $C=\frac 1{2a}$.
Taking like example your book

the constant $C$ is taken for future convenience.
Addendum: I have searched in the web and I have found this .pdf in Italian language. Excuse me. These notes have the same your approach (see the pag. 3) that I have highlighted and translated from Italian language.
where the constant has been named $1/2a$ for convenience.

