Projective invariance and the fifth coordinate

The paper considers the consequences of the principle of parametric (projective) invariance in the context of the general variational problem for stationary curves in a four-dimensional space-time. The necessity is shown of introducing a parameter along the trajectory as a fifth coordinate. The condition of cylindricality along it then acquires an obvious significance. The projective invariants are calculated for the trajectories of charged test particles in a gravielectromagnetic field. It is shown that one of these coincides with the density of the electromagnetic-field Lagrangian.