Relativistic particles with rigidity along light-like curves

We study actions in (2 + 1)-dimensions associated with null curves whose Lagrangians are arbitrary functions f on the curvature of the particle path, showing that null helices are always possible trajectories of the particles for every function f . The vector field P , obtained from the Euler-Lagrange equation, can be interpreted as the linear momentum of the particle since it is constant along the curve, which agrees with the conserved linear momentum law. The cases when f is constant or linear are completely solved and, by using Killing vector fields, we are able to integrate the Cartan equations in cylindrical coordinates around the linear momentum P . PACS numbers(s): 04.20.-q, 02.40.-k