Geometric Properties of Paths in Relativistic Lagrangian Mechanics

Considering an extension of the principle of covarience to the action along a path in relativistic Lagrangian mechanics, we motivate the use of geometric -- i.e. covariant and parameter invariant -- Lagrangian functions. We then study some properties of geometric Lagrangians, and introduce the notion of deviation of a path, which is a covariant measure of how much a path departs from a geodesic. Finally, we apply this notion of the twin paradox, and provide a rigorous resolution of it.