Continuous symmetry

A continuous symmetry belongs to a group whose parameters can vary continuously. Spontaneous breaking of such a symmetry can produce Goldstone bosons.

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.

Translation invariance means that shifting every position by the same constant vector leaves the system unchanged. It prevents an isolated interface or localized solution in a homogeneous infinite domain from having a preferred absolute position.