The Lagrange variety approach applied to frustrated classical wheels

The Lagrange variety approach introduced by Schmidt and Luban [J. Phys. A: Math. Gen. 36, 6351 (2003)] is applied to geometrically frustrated wheels (centered regular polygons). It is shown that the lowest energy configurations are planar or collinear. The latter one, characteristic for non-frustrated classical systems, is also observed in the presence of competing interactions in a well-determined range $(0, \alpha_c)$ of the energy function parameter $\alpha$. The ‘critical’ value $\alpha_c=1/4$ is universal, i.e., it does not depend on a system size. In this domain, the geometric frustration is present, but there is no non-trivial degeneracy.