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We formulate part I of a rigorous theory of ground states for classical,finite, Heisenberg spin systems. The main result is that all ground states canbe constructed from the eigenvectors of a real, symmetric matrix with entriescomprising the coupling constants of the spin system as well as certainLagrange parameters. The eigenvectors correspond to the unique maximum of theminimal eigenvalue considered as a function of the Lagrange parameters.However, there are rare cases where all ground states obtained in this way haveunphysical dimensions $M>3$ and the theory would have to be extended. Furtherresults concern the degree of additional degeneracy, additional to the trivialdegeneracy of ground states due to rotations or reflections. The theory isillustrated by a couple of elementary examples.

Theory of ground states for classical Heisenberg spin systems I