Magnetic angular-momentum invariant ($\mathbf{L}\cdot \mathbf{B}+\frac{q}{2}|\mathbf{r}\times \mathbf{B}{|}^2$)

For a particle obeying $m\dot{\mathbf{v}}=q\mathbf{v}\times \mathbf{B}$ in a constant uniform magnetic field, the displayed combination is conserved. The derivative of mechanical angular momentum projected onto the field is canceled by the quadratic position term. This is the rotational analogue of the distinction between mechanical and canonical momentum in an electromagnetic field.