An approach to the relativistic brachistochrone problem by sub-Riemannian geometry

We formulate a brachistochrone problem in Lorentzian geometry and we prove a variational principle valid for brachistochrones in stationary manifolds. This variational principle is stated in terms of geodesics in a suitable sub-Riemannian structure on M. Moreover, we prove the regularity of the solutions of our variational problem and we determine a differential equation satisfied by the brachistochrones, Some explicit examples are computed. (C) 1997 American Institute of Physics.