Axially symmetric isotropic-mass Lagrangian
= Axially symmetric isotropic-mass Lagrangian
If $L=|\dot q|^2/2-V(q)$ and $V$ depends only on $|q|^2$ and $(q\cdot n)^2$, then rotations about the axis $n$ leave $L$ invariant. The quantity supplied by <Noether theorem> is the axial angular momentum
$$
J_n=n\cdot(q\times\dot q).
$$
If $L$ also has no explicit time dependence, the energy $|\dot q|^2/2+V(q)$ is conserved independently.