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:
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?