Trouble with conservation of angular momentum in a system

[fluidistic]

Homework Statement

The system consists of a -fixed on the ground sphere- in which we attach a spring of natural length 0 and constant k at its north pole. At the end of the spring there's a particle of mass m. Gravity acts vertically downward. We stretch the spring and give an impulse to the particle in the $\hat \phi$ direction. The mass moves without friction over the surface of the sphere.
1)Write down the Lagrangian of the particle and the equations of motions using spherical coordinates.
2)What are the conserved quantities? Gives an expression of them in function of the coordinates and their time derivatives. Find the effective potential in function of theta only.
3)If the motion of the particle is constrained for 30 °< $\theta$ < 45°, find the values of the energy and angular momentum. Determine the range of the values for k for this situation to happen.

Homework Equations

L=T-V. A few more.

The Attempt at a Solution

$v^2=\dot r^2+r^2 \sin \phi \dot \theta ^2+r^2 \dot \phi ^2$. But since r is constant, $\dot r=0$. Thus $T=\frac{m}{2}(r^2 \dot \theta ^2 \sin ^2 \phi +r^2 \dot \phi ^2 )$.
The potential energy is the gravitational one ($mgr\cos \theta$) plus the one of the spring ($\frac{k\theta ^2 r^2}{2}$).
This gives me the Lagrangian $L=\frac{m}{2}(r^2 \dot \theta ^2 \sin ^2 \phi +r^2 \dot \phi ^2 )- \left ( mgr \cos \theta + \frac{k}{2} \theta ^2 r^2 \right )$.
However I do not see any of the 3 generalized coordinates r, theta and phi as cyclic.
One would expect the linear momentum in the direction of $\hat r$ to be constant and worth 0 and $\dot \phi$ to be a constant (different from 0) so that the angular momentum in the $\hat \phi$ direction would be conserved. But for that to happen, I shouldn't have gotten a $\phi$ term in my Lagrangian. I should have gotten $\phi$ as a cyclic coordinate for its generalized momentum (corresponding here to the angular momentum) to be conserved. However I do not get this. What am I doing wrong?!

 

[BigJDubs]

Also, I do not see how to find the effective potential in function of \theta only. For the 3rd question, I cannot start it until I manage to solve the first two.