= Magnetic axial angular-momentum invariant
{title2=$L_z+qB\rho^2/2$}
A charged particle in a uniform field $\boldsymbol B=B\boldsymbol e_z$ 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 $m(\dot{\boldsymbol r}\times\boldsymbol r)\cdot\boldsymbol B+e|\boldsymbol r\times\boldsymbol B|^2/2$. A radial restoring force has no axial torque.
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