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|-3Eijüj-3|u|-5(u·ú)Eijúj+const·(|u|-3(u·ú)ui-|u|2úi)=0