= Helical solenoid trajectory with zero incident angular momentum
Let a particle enter a uniform solenoid at transverse position $u_0=x_0+iy_0$ with zero radial velocity after an ideal fringe. Define signed $\Omega=qB/(2\gamma m)$ and initial field-free speed $v$. Its <Lorentz force> trajectory inside is
$$
u(t)=u_0\cos(\Omega t)e^{-i\Omega t}=\frac{u_0}{2}(1+e^{-2i\Omega t}),\qquad
z(t)=z_0+wt,\qquad w^2=v^2-\Omega^2|u_0|^2.
$$
The transverse motion is a circle about the guiding centre $u_0/2$ at cyclotron angular frequency $2|\Omega|$. Cylindrical radius is $|u_0\cos\Omega t|$; its apparent azimuthal jumps when the orbit crosses the axis are coordinate singularities. Forward entry requires $v>|\Omega u_0|$ in this ideal model.
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