To find the curve joining two points along which a particle sliding from rest under gravity travels from higher point to lower point in the least time
Let v be the velocity of particle when it is at p(x,y). the time taken by the particle to slide through a small distance ds along the curve = ds/v
Total time taken by the particle to slide from point
A to point B is t
12
= ∫ds/v
Now the total energy at A=(K.E)+(P.E)
T.E=T
A
+V
A
=0+0=0
Total energy at A= ½ mv 2 +(-mgy) , m is the mass and v is velocity

By conservation theorem, ½ mv 2 -mgy=0
 v 2 =2gy t
12
=
=
=
=
For t
1 to be minimum we have
-------(2)

=
=
=
=

Now f does not involve x explicitly.
 c ,where c is constant
Substituting values and solving the equations we get y(1+z 2 )=1/c 2 =b(say)
 y+yz 2 =b
 y 2 =b-y/y

Put y = bsin 2 φ and solving we get b-y=bcos 2 φ
Substituting in the above integral and solving we get b
But sin 2 φ=1-cos2φ.hence substituting we get b[φ- ½ (sin2 φ)]=x+c
1
When x=0,y=0 we get c
1
=0.so x = b[φ- ½ (sin2 φ)]

Also y=bsin 2 φ= b/2(1-cos2 φ).
Let b/2 =a and 2φ=θ.
Then the above equation becomes a(θ-sin θ)=x and a(1-cos θ)=y which is the parametric equation of cycloid.