A two-point boundary-value problem for gyroscopic systems in some Lorentzian manifolds

We study the dynamics of gyroscopic systems of relativistic type with multivalued action functionals. We assume that Lorentzian configuration manifolds have the structure of the twisted product. The solvability of the two-point boundary-value problem for such systems was proved earlier only in the case of a limited Lorentzian distance from the initial point to the final one. In this work we obtain a new existence theorem. According to this theorem, the specified distance to attainable points may be arbitrarily large. The result is applied to the dynamics of a charged test particle in the external space-time of the Reissner-Nordström black hole.