Mass Confined to Rotating Hoop with Moment of Inertia I3

[tphysicsb]

1. Homework Statement

A particle of mass can slide without friction on the inside of a small tube bent in a circle of radius r. The tube can rotate freely about the vertical axis, x3, and has a moment of inertia I3 about this axis.

Homework Equations

a) Derive the Hamiltonian for this system. I was able to determine the Hamiltonian to be.

b) List the cyclic coordinates and determine the conserved qty associated with each?

c) Write down Hamilton's equation of motion

The Attempt at a Solution

a) I was able to determine the Hamiltonian to be.
H= (Pθ)^2/2mr^2 + (P∅)^2/(2(I3+mr^2(sinθ)^2)) +mgrcosθ +c

b)I was able to determine the Lagrangian to be
L= 1/2mr^2 (theta dot)^2 + 1/2 mr^2(phi dot)^2(sinθ)^2 +1/2I3 (phi dot)^2 -mgrcosθ -c


Since the Lagrangian does not explicitly depend on phi(∅) which i believe this just ends up telling me that the conjugate momentum P∅ is constant ?

c) I used (q dot) = ∂H/∂pi and -pi = ∂H/∂qi

and found

(theta dot) = pθ/ mr2 = 1
p dot theta = mgrsinθ - mr^2 (phi dot)^2 sinθcosθ


P∅ = constant

(phi dot) = Pθ/ (I3 + mr^2(sinθ) ^2






I want to determine if I am on the right track with part b and part c. Specifically is their anything else to be be determined from part b other then the conjugate momentum (P∅ = constant) ??


Also Now that i determined the equations of motion in partC. What it is the next step to determine the equations of motion.

Any guidance you may have would be greatly appreciated

 

[Thaakisfox]

Cyclic coordinates just mean a coordinate upon which the lagrangian doesn't depend.

In c.) you determined the Hamilton equations of motion, so you don't need to go further.