homework and exercises - Working with a Routhian for a specific system

I asked a more general question earlier about the Routhian, but I'm still having trouble working with it. Here's my specific case. Given the following Lagrangian:

$$L=(1/2)m(\dot{r}^{2}+r^{2}\dot{\phi}^{2}sin^{2}\theta_{0})-mgrcos(\theta_{0}),$$

I obtained the corresponding Routhian:

$$R=(1/2)p_{\phi}\dot{\phi}-(1/2)m\dot{r}^{2}+mgrcos\theta_{0}.$$

I'm being asked to do the following: obtain the Hamilton equation of motion for $p_{\phi}$, show that it's constant in time and find its initial value, obtain the Lagrange equation of motion for $r$ from the Routhian, and obtain the Hamiltonian

When I try to do the first part (obtain the Hamilton e.o.m.) I get $∂R/∂p_{\phi}=(1/2)\dot{\phi}$ which doesn't show that it's constant in time or that angular momentum is conserved, which I think is what we're trying to show here. What am I doing wrong? Also, how can I obtain the Lagrange e.o.m. for $r$ using the Routhian? Do I just take

$$\frac{d}{dt}\frac{∂R}{∂\dot{r}}-\frac{∂R}{∂r}=0,$$

since $r$ appears in the Routhian and plug in $r=r_{0}$ to find the initial value? Similarly, how do I go about obtaining the Hamiltonian from the Routhian?