Using Lagrange to solve rotating parabolic motion and equilibrium

[relativespeak]

Homework Statement

Consider a bead of mass m sliding without friction on a wire that is bent in the shape of a parabola and is being spun with constant angular velocity w about its vertical axis. Use cylindrical polar coordinates and let the equation of the parabola be z = kp2. Write down the Lagrangian in terms of p as the generalized coordinate. Find the equation of motion of the bead and determine whether there are positions of equilibrium, that is, values of p at which the bead can remain fixed, without sliding up or down the spinning wire. Discuss the stability of any equilibrium positions you find.

Homework Equations

L=E-U
v1=pw
v2=dp/dt
v3=2kp(dp/dt)
U=mgkp^2
E=mv^2/2

The Attempt at a Solution

I found the equation of motion using the lagrangian, which matched the answer in the back of the book. I just don't know how to use Lagrance to find equilibrium, and I've tried several things such as setting d²p/dt²=0 and d²U/dp²=0.The equation of motion is: (g=accel. due to gravity)

pw²-4k²p(dp/dt)²-2kgp=(d²p/dt²)(1+4k²p²

 

[voko]

If the bead remains fixed, what can be said about the derivatives of its polar radius?

 

[relativespeak]

dp/dt=0, d²p/dt²=0

so equil. occurs when w²=2kg, or when p=0. Is that right? And then how can I find which is stable/unstable?

 

[voko]

Have you studied any equilibrium conditions/criteria?