Analytical Parameterization of Rotors and Proof of a Goldberg Conjecture by Optimal Control Theory

Curves which can be rotated freely in an n-gon (that is, an regular polygon with n sides) so that they always remain in contact with every side of the n-gon are called rotors. Using optimal control theory, we prove that the rotor with minimal area consists of a finite union of arcs of circles. Moreover, the radii of these arcs are exactly the distances of the diagonals of the n-gon from the parallel sides. Finally, using the extension of Noether’s theorem to optimal control (as performed in [D. F. M. Torres, WSEAS Trans. Math., 3 (2004), pp. 620–624]), we show that a minimizer is necessarily a regular rotor, which proves a conjecture formulated in 1957 by Goldberg (see [M. Golberg, Amer. Math. Monthly, 64 (1957), pp. 71–78]).