A general variational formulation for relativistic mechanics based on fundamentals of differential geometry

The first part of this article develops a variational formulation for relativistic mechanics. The results are established through standard tools of variational analysis and differential geometry. The novelty here is that the main motion manifold has a $n+1$ dimensional range. It is worth emphasizing in a first approximation we have neglected the self-interaction energy part. In its second part, this article develops some formalism concerning the causal structure in a general space-time manifold. Finally, the last article section presents a result concerning the existence of a generalized solution for the world sheet manifold variational formulation.