Conformal Mechanics of Planar Curves

Abstract

It is possible to associate curvature-dependent Möbius invariant energies with planar curves. They are of particular interest because their tension-free stationary states, lacking a length scale, form self-similar curves. As such one would expect these energies to show up in self-similar growth processes. The simplest among them is the conformal arc-length. Its tension-free states are logarithmic spirals characterized by the rate of growth; in general its stationary states have constant conformal curvature: but any such state can be generated from a logarithmic spirals by applying an appropriate Möbius transformation to it. In the process, tension is introduced and with it a length scale. Generically these Möbius descendants form double spirals, characterized up to similarity by a second parameter, the distance between the two poles. In this paper, we will show how this invariant system can be approached from a mechanical point of view. Central to this approach is the identification of the four currents conserved along stationary curves: the tension and torque associated with Euclidean invariance; in addition scalar and vector currents reflect the invariance under scaling and special conformal transformations respectively. It will be shown that the conformal Casimir invariant, proportional to the conformal curvature, can be decomposed in terms of these currents. This approach permits the familiar geometrical properties of logarithmic spirals to be reinterpreted in mechanical terms, while suggesting new ones. In logarithmic spirals, not just the tension but also the special conformal current vanish with respect to a reference frame centered on the spiral apex. The torque M and the scaling current S are constrained to satisfy 4M S = 1 in these states. The tension generated in a double spiral is evaluated. While it is, as expected, inversely proportional to the distance between its poles, it is not directed along the line connecting these two points; the non-vanishing angle it makes with this direction is the angle defining the logarithmic spiral, independent of the inter-polar distance. An explicit construction of these states using the conserved currents will be presented. In higher order Möbius invariant theories, logarithmic spirals generally get decorated by self-similar internal structure.

Conformal mechanics of space curves

“…Tension-free curves are necessarily in equilibrium. In contrast to the planar reduction of this problem, however, it is not obvious if every equilibrium state is conformally equivalent to a tension-free state [17].…”

“…( 2). Contrast the one parameter planar logarithmic spirals described in reference [17] with the zoo of self-similar similar space curves described in reference [18]. Unlike logarithmic spirals, they generally exhibit internal structure associated with non-vanishing torsion.…”

“…The approach we adopt to examine the behavior of H under small deformations, X → X + δX, will be the method of auxiliary variables, developed originally to examine surfaces [22] (see also [9] for a recent review, tailored to the two-dimensional bending energy). This provides an efficient way to identify conserved currents associated with the symmetries of any energy H depending on geometric degrees of freedom [22,17]. While originally developed for energies quadratic in curvature, there is no obstacle to considering energies involving higher derivatives (see, for example, [32] or [33], a factor of two and a sign error in [32] were corrected discreetly in the more general treatment in [33]).…”

“…It is significant that H possesses logarithmic spirals as planar critical points [16]. The planar limit has been examined from a mechanical point of view in [17]).…”

“…The language of Euler elasticity is now appropriate, extended to accommodate the additional symmetry. This route builds on the approach adopted, in the context of planar curves, in reference [17].…”