Brachistochrone is Cycloid/Proof 1

Proof

 * Brachistochrone.png

Recall from the Snell-Descartes Law:
 * $\dfrac {\sin \alpha_1} {v_1} = \dfrac {\sin \alpha_2} {v_2}$

Here, we invoke a generalization of the Snell-Descartes Law.

This is justified, as we are attempting to demonstrate the curve that takes the smallest time.

Thus we have $\dfrac {\sin \alpha} v = k$, where $k$ is some constant.

By the Principle of Conservation of Energy, the speed of the bead at a particular height is determined by its loss in potential energy in getting there.

Thus, at the point $\tuple {x, y}$, we have:

We have:

Combining all the above equations, we get:

where $c$ is another (more convenient) constant.

This is the differential equation which defines the brachistochrone.

Now we solve it:

Now we introduce a change of variable:

Let $\sqrt {\dfrac y {c - y} } = \tan \phi$

Thus:

Also:

Thus:

As the curve goes through the origin, we have $x = y = 0$ when $\phi = 0$ and so $c_1 = 0$.

Now we can look again at our expression for $y$:

To simplify the constants, we can substitute $a = c / 2$ and $\theta = 2 \phi$, and thus we get:

which are the parametric equations of the cycloid.