Central Force using Lagrangian

I am working through the book "Emmy Noether's Wonderful Theorem" by Dwight Neuenschwander and have a problem with his central force example.

He gives the particle velocity in spherical coordinates as:

so the Lagrangian

He then says that using the Euler-Lagrange equations we find P_theta and p_phi are constant. But using the Euler-Lagrange equation gives:

and

So it seems to me that P_theta cannot be constant. Even though based on the physics it must be? What am I missing here?

Re: Central Force using Lagrangian

Thinking about this problem, and expanding the Lagrangian equation for theta, the RHS is cyclic while the LHS is not. I'm convinced this can only happen if both sides are constant, and this constant can only be zero, since the RHS is zero for theta = pi/2