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:

$\displaystyle \underline v = \dot r \hat r + r \dot \theta \hat \theta + r \dot \phi sin \theta \hat \phi$

so the Lagrangian

$\displaystyle L = \frac 12 m \underline v . \underline v - U(r) = \frac 12 m (\dot r^2+r^2 \dot \theta ^2 + r \dot \phi ^2 sin^2 \theta)-U(r)$

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:

$\displaystyle \dot p_\theta=\frac d{dt}\frac{\partial L}{\partial \dot \theta}=\frac d{dt}mr^2\dot\theta=\frac{\partial L}{\partial \theta} \neq 0$

and

$\displaystyle \dot p_\phi=\frac d{dt}\frac{\partial L}{\partial \dot \phi}=\frac d{dt}mr^2\dot\phi sin^2 \theta=\frac{\partial L}{\partial \phi} = 0$

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?