We review some results concerning the convexity of the boundary of a standard domain M = D × R contained in a standard stationary spacetime L = S × R showing that this notion can be characterized by means of Riemannian conditions involving a potential and a one-form on S, both linked to the coefficients of the metric. A natural application, obtained by variational methods, is the ex- istence of stationary geodesics in M connecting a point to a stationary line and having fixed Lorentzian length.

We prove an existence result for trajectories of classical particles accelerated by a potential and a magnetic field on a non–complete Riemannian manifold M . Both the potential and the magnetic field may be not bounded and have critical growth. We state a suitable convexity assumption involving the magnetic field in order to prove that the support of each trajectory is entirely contained in M .