Ground states of classical magnetic dipole rings

We investigate the two well-known ground states of rings of $N$ classical magnetic dipoles that are given by clockwise or anti-clockwise spin orientations tangent to the circle encompassing the dipole ring. In particular, we formulate a rigorous proof of the ground state property of the states in question. The problem can be reduced to the determination of the lowest eigenvalue of a $3N\times 3N$ matrix ${\mathbf J}$. We show that all eigenvalues of ${\mathbf J}$ can be analytically calculated and, at least for $N=3,\ldots,8$, the lowest one can be directly determined. The main part of the paper is devoted to the completion of the proof for $N\ge 9$ based on various estimates and case distinctions. We also discuss the question to what extent computer-algebraic results should be allowed to contribute to a mathematical proof.