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

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