Conformal Mechanics of Planar Curves

Abstract

Self-similar curves arise naturally as the tension-free equilibrium states of conformally invariant bending energies. The simplest example is the M\"obius invariant conformal arc-length on planar curves, dependent on the Frenet curvature $\kappa$ through its first derivative with respect to arc-length. There are four conserved currents associated with this invariance: the tension and torque associated with Euclidean invariance, as well as scalar and vector currents reflecting invariance under scaling and special conformal transformations respectively. If the tension vanishes, all equilibrium states are self-similar: in the case of conformal arc-length, these are logarithmic spirals with no internal structure. More generally, the tension-free states are logarithmic spirals decorated with a repeating self-similar internal structure. Here it will be shown how the conservation laws can be used to construct these curves, while also endowing their geometry with a mechanical interpretation. The scaling current and the torque together provide a scale-invariant ode for the dimensionless variable $\kappa'/\kappa^2$, which captures the internal structure of the spiral. For conformal arc-length it is constant. In tension-free states, the special conformal current vanishes. Its projections along orthogonal directions determine directly the distance from the spiral apex locally in terms of the curvature. The quadratic Casimir invariant of the M\"obius group can be cast in terms of the four currents, none of which itself is invariant. For conformal arc-length, this is identified as the conformal curvature (the Schwarzian derivative of the Frenet curvature); it is constant along equilibrium curves.