Homework Help: Precession of a gyroscope

1. Mar 12, 2013

aaaa202

Its a long time since I have done any mechanics and today I thought I would like to work out the equations of motion for a gyroscope. So I tried to put up a Lagrangian for a simple system (on the picture) of a rod of length L that is free to rotate about a pivot which has a flywheel (the red disk) which is free to rotate around about the axis of the rod.
The lagrangian I foundd was simply: L = ½Iaθ'22 + ½Ibψ'2+mgcos(θ)L

But as you can see this Lagrangian fails as the equations of motion gives no dependence of ψ for the motion about the angle θ, which I would assume it should since the angular momentum of the flywheel about the axis of the rod dictates how the rod precesses. What am I doing wrong? Is it because ψ is not independent of θ? And what is the correct way of doing this problem, I havent done this kind of mechanics in a long time, so sorry if Im way off.

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