Summary: The variational principle and the corresponding differential equation for geodesic circles in two dimensional (pseudo)-Riemannian space are discovered. The relationship with the physical notion of uniformly accelerated relativistic particles is emphasized. The known form of spin-curvature interaction emerges due to the presence of second order derivatives in the expression for the Lagrange function. The variational equation itself reduces to the unique invariant variational equation of constant Frenet curvature in two dimensional (pseudo)-Euclidean geometry.