The question is to find the shape of the curve down whcih a bead sliding from rest and accelerated by gravity will slip(without friction) from one point to another in the least time.
So I proceeded with the kinematics equations, $ S = ut + 1/2 at^ 2 $, from whcih we finally get the time required to travel from a point P1 to P2 to be,
$ t= \int_{P1}^{P2} (2S/a)^{1/2} $ ,
where S is given by $ (1 +(dy/dx)^2)^{1/2} \,dx $,
The t becomes,
$ t = \int_{P1}^{P2} ((2(1+(dy/dx)^2)^{1/2} \,dx)/a)^{1/2} $,
Then I want to minimize the above function which is a function of $y,x$ and $dy/dx$ finally to get the equation of a cycloid.
We can do it by the method of variations, but I just tried to integrate the function directly , but integral involves $(dx)^{1/2}$, so I am not able to proceed furthur. Can anyone help me to give an idea to solve the integral involving fractional power of differential $dx$ .
Thanks in advance.