Relativistic particles with rigidity along light-like curves AngelFerr´andez 1 , Angel Gim´ enez 2 and Pascual Lucas 3 Departamento de Matem´ aticas, Universidad de Murcia 30100 Espinardo, Murcia, Spain Abstract 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 Keywords: lightlike worldline, spinning massless and massive particles, moduli spaces of solutions. 1 Introduction For the past fifteen years, many interesting papers concerning Lagrangians describing spinning particles have been published (see e.g. [1]–[18] and references therein). In the general situation, as it is well known, one has to provide the classical model with the extra bosonic variables. To this end, an interesting hypothesis deals with Lagrangians on higher geometrical invariants to supply those extra degrees of freedom. This approach has the interesting point of view that the spinning degrees of freedom are encoded in the geometry of its world trajectories. The Poincar ´ U and invariance requirements imply that an admissible Lagrangian density F must depends on the extrinsic curvatures of curves in the background gravitational field. In particular, the Lagrangians depending on the first and second curvatures have been intensively studied in the late eighties and in the nineties. At the beginning those systems were studied as toy models of rigid strings and (2+1)-dimensional field theories with the Chern-Simons term but shortly after, mainly due to the papers by Plyushchay, those systems are of independent interest. The actions considered before are defined on non-isotropic curves (spacelike or time- like), but on (d + 1)-spacetimes one can also consider actions defined on null (lightlike) 1 aferr@um.es 2 agpastor@um.es 3 plucas@um.es, corresponding author. FAX number: +34-68-364182 1