From Euler's elastica to the mKdV hierarchy, through the Faber polynomials

TL;DR

This paper explores the connection between Euler's elastica, the mKdV hierarchy, and Faber polynomials by analyzing geometric constraints on plane loops and their invariants.

Contribution

It introduces a novel link between elastica constraints and the mKdV hierarchy via Faber polynomials, expanding understanding of geometric flows and invariants.

Findings

Derivation of the mKdV hierarchy from elastica conditions

Identification of curvature constraints as Faber polynomial coefficients

Establishment of a geometric interpretation of Faber polynomials

Abstract

The modified Korteweg-de Vries hierarchy (mKdV) is derived by imposing isometry and isoenergy conditions on a moduli space of plane loops. The conditions are compared to the constraints that define Euler's elastica. Moreover, the conditions are shown to be constraints on the curvature and other invariants of the loops which appear as coefficients of the generating function for the Faber polynomials.