Relativistic particles and the geometry of 4-D null curves (Q2456453)

In the last years, many interesting papers concerning Lagrangians describing spinning particles have been published [see \textit{Y. A. Kuznetsov} and \textit{M. S. Plyushchay}, J. Math. Phys. 35, No. 6, 2772--2784 (1994; Zbl 0808.70014); \textit{M. S. Plyushchay}, Modern Phys. Lett. A 4, No. 9, 837--847 (1989)]. In this paper the authors consider actions in \((d+1)\) dimensions whose Lagrangians are linear functions on the curvature of the particle path. Then the action \[ {\mathcal{L}}:\Lambda\Rightarrow R, \] given by \[ {\mathcal{L}}(\nu) = \int_{\nu}(k_1\mu + \lambda)\,d\sigma, \] is considered, where \(\mu\) and \(\lambda\) both are constant and \(k_1\) stands for the first curvature of the null curve. The equations of motion for these Lagrangians are completely given in \((d+1)\)-background gravitational fields. The authors solve the motion equations and get the null wordlines of the relativistic particles in cylindrical coordinates around a certain plane \(\pi\) according to whether \(\pi\) in nonnull or null. In the last part of the paper a complete description of critical curves is given, by using cylindrical coordinates of the Lagrangian \({\mathcal{L}}(\nu)\).