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?