The functional definition of geodesics

In this paper a functional definition of geodesics is introduced which allows to generalize the notion of a geodesic from smooth to topological manifolds. It is shown that in the smooth case the new definition coincides with the classical definition of geodesics of a linear connection. If the smoothness is not required, it is shown by an example that the new definition includes geodesics of non-linear, homogeneous connections. Moreover, an example of generalized geodesics which does not arise from any connection is presented here.