Helical symmetry ($(\theta ,z)\mapsto (\theta +ϵ,z+bϵ)$)

A helical symmetry combines an axial rotation and a proportional axial translation. A helix with $z=b\theta$ is unchanged as a set by this transformation. If a mechanical Lagrangian is invariant, the Noether conserved quantity for a mechanical point symmetry is $L_z+b{P}_z$, where the two momenta include all moving components. Pure axial rotation and pure axial translation need not separately be symmetries. The helical symmetry of a potential is a related special case in which the constraint is replaced by an invariant potential.^[ref]

If zero conserved helical momentum reduces to $r^2\dot{\theta }+b\dot{z}=0$, a closed transverse projection $C$ has axial displacement $\Delta z=-b^{-1}{\oint }_C(xdy-ydx)=-2A/b$. Here $b\ne 0$ 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.