Geometrical Particle Models In Constant Curvature Space Forms On Null Curves

In this paper we consider the mechanical system which is the case of the action arbitrary depending on Cartan curvature and the motion equations for this Lagrangian is rigorously obtained in 4-dimensional Lorentz space forms. We want to find three Killing fields along the critical curves and get some conservation laws. But when it is hard to do, we will turn to 3-dimension constant curvature Lorentz space. We construct a suitable coordinate system by using these Killing fields and express the critical curves by quadratures.