In this paper we introduce a reference along a null curve in an n-dimensional Lorentzian space with the minimum number of curvatures. That reference generalizes the reference of Bonnor for null curves in Minkowski space–time and it is called the Cartan frame of the curve. The associated curvature functions are called the Cartan curvatures of the curve. We characterize the null helices (that is, null curves with constant Cartan curvatures) in n-dimensional Lorentzian space forms and we obtain a complete classification of them in low dimensions.

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The simplest (2+1)-dimensional mechanical systems associated with light-like curves, already studied by Nersessian and Ramos, are reconsidered. The action is linear in the curvature of the particle path and the moduli spaces of solutions are completely exhibited in 3-dimensional Minkowski background, even when the action is not proportional to the pseudo-arc length of the trajectory.Comment: Corrected typos and english flaws. Final version accepted in Phys. Lett.

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We study actions in (d + 1)-dimensions associated with null curves, mainly when d = 3, whose Lagrangian is a linear function on the curvature of the particle path, showing that null helices are always possible trajectories of the particles. We find Killing vector fields along critical curves of the action which correspond to the linear and the angular momenta of the particle. They provide two constants of the motion which can be interpreted in terms of the mass and the spin of the system. Moreover, we are able to integrate both the Euler-Lagrange equations and the Cartan equations in cylindrical coordinates around a certain plane.

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We obtain a Lancret-type theorem for null generalized helices in Lorentz-Minkowski spaces. In the 3-dimensional space we get that the only null generalized helices are the ordinary null helices. In the 5-dimensional space, we distinguish between null generalized helices with non-null or null axis and in both cases we solve the natural equations problem.