Magnetic axial angular-momentum invariant ($L_z+qB{\rho }^2/2$)

A charged particle in a uniform field $\mathbf{B}=B{\mathbf{e}}_z$ and an axially symmetric potential conserves the displayed canonical angular momentum, with $\rho$ the distance from the field axis and $q$ its charge. It follows directly by differentiating the expression and using the Lorentz force. For charge $q=-e$, the equivalent invariant is $m(\dot{\mathbf{r}}\times \mathbf{r})\cdot \mathbf{B}+e|\mathbf{r}\times \mathbf{B}{|}^2/2$. A radial restoring force has no axial torque.