Curve of fastest descent

In 1696, Johann Bernoulli proposed the problem of finding the curve of fastest descent: given a point $A$ and a lower point $B$ which is not directly below $A$, find the curve lying on the plane between the two points on which a bead slides frictionlessly under the influence of a uniform gravitational field in the shortest time. It was then famously solved by Issac Newton in 1697. See this wiki page.

The curve is called Brachistochrone curve, which is a cycloid. 

I give my solution below.

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Let the curve we are seeking be $L$. We flip it upside-down s.t. the bead has velocity $\sqrt{2gy}$ at height $y$. Divide $L$ into $n$ very small segments of vertical height $\Delta y$, horizontal lengths $\Delta x_1,\Delta x_2,\dots,\Delta x_n$, and heights $y_1,y_2,\dots,y_n$. Then the following is minimized 

$$\sum_{i=1}^n\frac{\sqrt{\Delta y_i^2+\Delta x_i^2}}{\sqrt{y_i}}$$

such that $\sum_{i=1}^n\Delta x_i$ is the horizontal difference of $A$ and $B$. Langrange multiplier implies that

$$y\left(1+\left(\frac{dy}{dx}\right)^2\right)=2R$$

throughout $L$ for some positive constant $R$.

Let $\phi$ be the angle between the tangent line of $P(x,y)$ on $L$ and the $x$-axis. Then

$$\frac{y}{\cos^2\phi}=2R$$ throughout $L$, or

$$y=R\left(1+\cos2\phi\right).$$

We have $\frac{dy}{d\phi}=-2R\sin2\phi$, so let $\frac{dx}{d\phi}=-2R\frac{\sin2\phi}{\tan\phi}=2R(1+\cos2\phi)$. Then

$$x=2R\left(\phi+\frac{1}{2}\sin2\phi\right)+C.$$

Define $\theta:=2\phi-\pi$, we have

$$x=R\left(\theta-\sin\theta\right)+C,$$

$$y=R\left(1-\cos\theta\right).$$

So $L$ is a cycloid. Note that for $y=0$ we have $\phi=\pi/2$ and $\theta=0$. This makes sense, because no horizontal movement should be allocated to the minimum velocity.

How to solve $R$ and $\theta_0$ where $A=(0,0)$ and $B=\left(R\left(\theta_0-\sin\theta_0\right), R\left(1-\cos\theta_0\right)\right)$? Note that the mapping

$$f(\theta)=\frac{\theta-\sin\theta}{1-\cos\theta}$$

is bijective for $\theta\in(0,2\pi)$, so $\theta_0$ can be uniquely determined by the ratio of two coordinates of $B$, after which $R$ can be determined as well.