Analytical Mechanics: Proving Equilibrium at Bottom of Bowl

[noor]

Homework Statement
A point particle of mass m is confined to the frictionless surface of a spherical bowl. There are 2 degrees of freedom.
1. Prove that the equilibrium point is the bottom of the bowl. ,
2.Does the bowl need to be exactly spherical for this to be true ?
3. Near the bottom of the bowl what is the most general form possible for the shape of the bowl in order to maintain the stability of the equilibrium point at the bottom?
I found this problem in analytical mechanics book(Hand and Finch) and was trying to solve it.

Homework Equations

The Attempt at a Solution

In part one i used the p.e to find the eq point.
im not sure about part 2&3 please !

 

[jedishrfu]

For part 2 you could consider various surfaces of revolution like a cone or a parabolic shape and from there show whether a spherical surface is necessary.