**Variational principle for the 2-dimensional concircular geometry**

**Abstract:**

The concircular geometry deals with curves of constant first
curvature and of zero second curvature. The corresponding third-order
differential equation of such a path coincides with the equation
of uniformly accelerated test particle in General Relativity. We
extend to the case of general 2-dimensional
pseudo-Riemannian geometry the following result: there exists the
unique Lorentz-invariant variational equation of the third order, the
integral paths of which have constant curvature and include the usual
straight geodesics:

|u|^{-3}E_{ij}ü^{j}-3|u|^{-5}(u·ú)E_{ij}ú^{j}+const·(|u|^{-3}(u·ú)u_{i}-|u|^{2}ú_{i})=0