= Signed-area displacement under a helical constraint
{title2=$\Delta z=-2A/b$}
If zero conserved helical momentum reduces to $r^2\dot\theta+b\dot z=0$, a closed transverse projection $C$ has axial displacement $\Delta z=-b^{-1}\oint_C(x\,dy-y\,dx)=-2A/b$. Here $b\ne0$ and $A$ is the <signed area>, with winding multiplicity for self-intersections. This is a path-dependent displacement even when the transverse coordinates return to their initial values. It follows from the momentum constraint and <Green's theorem>; it is not a prediction that every chosen closed curve is dynamically realizable.
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