Canonical angular momentum across a solenoid fringe

For an axisymmetric magnetic vector potential, the conserved angular canonical momentum is $p_{\theta }=\gamma m{\rho }^2\dot{\theta }+q\rho A_{\theta }$. A particle arriving parallel to a solenoid axis from a field-free region has $p_{\theta }=0$. In a uniform field, $A_{\theta }=B\rho /2$ and $\dot{\theta }=-qB/(2\gamma m)$ wherever $\rho \ne 0$. The mechanical angular momentum changes through the fringe while the canonical quantity remains constant. Static magnetic fields also conserve $\gamma m{c}^2$; the azimuthal kick therefore reduces axial kinetic motion rather than increasing total speed.