Lagrangian classical relativistic mechanics of a system of directly interacting particles. I

General formulation of the one-time Lagrangian relativistic classical description of $N$ directly interacting particles is developed. A representation of a continuous transformation group of the Minkowski space (in particular, Poincare group) by the Lie–Backlund tangent transformations of the configuration space of the system is constructed. By means of this representation the system of linear differential equations expressing the Poincare invariance of the Lagrangian formalism is obtained and corresponding restrictions on the form of the generalised Lagrangian are studied. The exact relativistic description of the interaction is shown to demand the dependence of the Lagrangian on the infinite order derivatives. The results will be used in the second part of the work for finding a general form of the approximate relativistic interaction Lagrangian and for working out the method of constructing the Poincare invariant Newton type equations of motion and their first integrals.