Lagrangian problem: Ball oscillating in spherical bowl

[Lengalicious]

Homework Statement

Consider a solid sphere of radius r to be placed at the bottom of a spherical bowl radius R, after the ball is given a push it oscillates about the bottom. By using the Lagrangian approach find the period of oscillation.

Homework Equations

The Attempt at a Solution

Ok so this is as far as i get, not sure if what I am doing is correct:
General coords (∅,x,y)

T = 1/2*m*(xdot²+ydot²) + 1/2*I*∅dot²
T = 1/2*m*xdot²+1/2*m*ydot²+(mr²∅²)/5 (Since Inertia of Solid Sphere is = 2/5*m*r²)
V = mgy

L = 1/2*m*xdot²+1/2*m*ydot²+(mr²∅²)/5 - mgy

∂L/∂∅dot = 2/5*m*r²*∅dot
∂L/∂∅ = 0

∂L/∂xdot = m*xdot
∂L/∂x = 0

∂L/∂ydot = m*ydot
∂L/∂y = -mg

d(2/5*m*r²*∅dot)/dt = 0
d(m*xdot)/dt = 0
d(m*ydot)/dt = -mg

Up to here I am not sure whether I've been doing the right thing, and also, not quite sure how this helps me find the period of oscillation?

EDIT: Sigh, I know my equations of motion are wrong because i should be getting something similar to the DE for an SHO. Could really use some help.

 

[clamtrox]

There are some conditions you need to take into account: because bowl is spherical, you have a relation between x- and y-coordinates, and because the ball is rolling, there is a relation between phi dot and x dot and y dot.

 

[Lengalicious]

Ok, but did I use the correct coordinate system? Or should it be phi and theta, and once I have the constraints how do I encorporate them into the equation of motion?