# Mass confined to a rotating hoop with a moment of inertia I3 about the vertical axis

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