Brachistochrone is Cycloid

Curve
Let a point $$A$$ be joined by a wire to a lower point $$B$$.

Suppose we are allowed to bend the wire into whatever shape we want.

Suppose that a bead is allowed to slide down without friction from $$A$$ to $$B$$.

The shape of the wire so that the bead takes least time to descend from $$A$$ to $$B$$ is a cycloid.

Proof

 * Brachistochrone.png

We invoke a generalization of Snell's Law.

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

Thus we have $$\frac {\sin \alpha} v = k$$, 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 $$\left({x, y}\right)$$, we have:
 * $$v = \sqrt {2 g y}$$

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:
 * $$\sqrt {\frac y {c - y}} = \tan \phi$$

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.