Lagrange formalism of point-masses

We prove by symmetry properties that the Lagrangian of a free point-mass is a quadratic function of the speed in the non-relativistic case, and that the action of the free point-mass between two spacetime points is the proper time passed in the relativistic case. These well known facts are proved in a mathematically rigorous way with a frame independent treatment based on spacetime models introduced by Matolcsi. The arguments show that these results are not obvious at all, some common beliefs can be refuted by explicit counterexamples. In our treatment the similarity of non-relativistic and relativistic cases is apparent.