Cycloid pendulum (Huygens)

[alejandrito29]

"Huygens' ingenious idea, which he put into practice, was to vary the eective length of the pendulum by allowing its cord to wrap partially around an obstruction as it swings"

the coordinates of cycloid are:

$$x=a(\theta -\sin \theta)$$
$$y=a(\cos \theta +1)$$

why in somes articles, they use the coordinates x,y for calculate de Lagrangian ?, when the unknown quantity is the period. My question is because, the coordinates of mass of pendulum not are x and y (this are the coordinates of cycloid).

for example in this links use x and y.

http://forum.lawebdefisica.com/threads/1873-Péndulo-cicloidal

 

[Simon Bridge]

I'm not sure what you are asking about:
People are free to use whatever coordinate system they want.
Usually there is some advantage to the choice.

In your link - the choice of coordinates is not due to the website you link to, but to the people discussing the question. That is their personal choice. Maybe those people just find Cartesian coordinates comfortable?

Have you tried constructing the Lagrangian in another way?