Motion along cycloidal curves

The following simulation try to illustrate the properties of cyclodial curves.

$\theta=\omega t$ , $x=r(\theta-\sin\theta), y=r(1+\cos\theta)$
so $x'=r(1-\cos\theta), y'=r\sin\theta$
$x'^2+y'^2=2r^2(1-\cos\theta)$
From conservation of energy: $\tfrac{1}{2}mv^2=mgy$ , $v=\tfrac{ds}{dt}=\sqrt{2gy}$
$dt=\frac{ds}{v}=\frac{\sqrt{dx^2+dy^2}d\theta}{\sqrt{2gy}}=\frac{\sqrt{2r^2(1-\cos\theta)}}{\sqrt{2gr(1-\cos\theta)}}=\sqrt{\frac{r}{g}}d\theta$ , so $T_{1/4}=\sqrt{\frac{r}{g}}\pi$

However, from an intermediate point $\theta_0$ , $v=\frac{ds}{dt}=\sqrt{2g(y-y_0)}$
So $T=\int_{\theta_0}^{\pi}\sqrt{\frac{2r^2(1-\cos\theta)}{2gr(\cos\theta_0-\cos\theta)}}d\theta$

with $\sin\frac{x}{2}=\sqrt{\frac{1-\cos x}{2}}, \cos\frac{x}{2}=\sqrt{\frac{1-\cos x}{2}}$

$T=\sqrt{r}{g}\int {\theta_0}^{\pi} \sqrt{\frac{\sin\tfrac{\theta}{2}d\theta}{\cos^2\tfrac{\theta_0}{2}-\cos^2\tfrac{\theta}{2}}}$
define $u=\frac{\cos\tfrac{\theta}{2}}{\cos\tfrac{\theta_0}{2}}, du=\frac{\sin\frac{\theta}{2}d\theta}{2\cos\tfrac{\theta_0}{2}}$

$T=\sqrt{r}{g}\int_1^0 \frac{du}{\sqrt{1-u^2}}=2\sqrt{r}{g}\sin^{-1}u|_0^1=\sqrt{r}{g} \pi$
So the amount of time is the same from any point.

1. A cycloid is the curve defined by the path of a point on the edge of circular wheel as the wheel rolls along a straight line.
2. Cycloidal curves is the curve of fastest descent under gravity
3. the period of a ball rolling back and forth inside this curve does not depend on the ball’s starting position.

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