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

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