Classical MechanicsI M.Sc(Maths)

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 timeLet 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/vTotal time taken by the particle to slide from point A to point B is t12 = ∫ds/v

Now the total energy at A=(K.E)+(P.E)T.E=TA+VA=0+0=0

Total energy at A= ½ mv2+(-mgy) , m is the mass and v is velocity

By conservation theorem, ½ mv2-mgy=0v2=2gyt12= =

= =

For t1 to be minimum we have

-------(2)

=

=

=

=

-----(3)

Now f does not involve x explicitly.

c ,where c is constant

Substituting values and solving the equations we get y(1+z2)=1/c2=b(say)

y+yz2=by2=b-y/y

Put y = bsin2φ and solving we get b-y=bcos2φSubstituting in the above integral and solving we get

b

But sin2 φ=1-cos2φ.hence substituting we get b[φ- ½ (sin2 φ)]=x+c1

When x=0,y=0 we get c1=0.so x = b[φ- ½ (sin2 φ)]

Also y=bsin2φ= 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.