= 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 $\rho\neq0$. The mechanical angular momentum changes through the fringe while the canonical quantity remains constant. Static magnetic fields also conserve $\gamma mc^2$; the azimuthal kick therefore reduces axial kinetic motion rather than increasing total speed.
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