Conformal mechanics of space curves

Abstract

Any conformally invariant energy associated with a curve possesses tension-free equilibrium states which are self-similar. When this energy is the three dimensional conformal arc-length, these states are the natural spatial generalizations of planar logarithmic spirals. In this paper, a geometric framework is developed to construct these states explicitly using the conservation laws associated with the symmetry. The tension along a curve, conserved in equilibrium, is first constructed. While the tension itself is not invariant, the statement of its conservation is. By projecting the conservation laws along the two orthogonal invariant normal directions, the Euler-Lagrange equations are reproduced in a manifestly conformally invariant form involving the conformal curvature and torsion. The conserved torque, as well as scaling and special conformal currents implied by the symmetry are constructed explicitly. The special conformal current vanished with respect to an appropriate origin in all tension-free states. A sketch is provided of how self-similar spirals describing tension-free states can be constructed by integrating the conservation laws. The details will be provided in a companion paper, arXiv:1904.06876.