For a given manifold M, modelled on a Banach space B, second order differential equation provides an alternative way to study geometric structures on M. Firstly for every connection r on M we associate a second order differential equation S in a way that the r-geodesics are geodesics with respect to S. In a further step despite of natural diffculties with non-Banach modelled manifolds, and even spaces, we generalize these results to a wide class of Frechet manifolds. More precisely we show that for a Frechet manifold M, which can be considered as projective limit of Banach manifolds, for a given initial value there exists a unique geodesic. As an interesting result we propose two criterions to generalize the concept of completeness for a wide class of Frechet manifolds. The last part of the paper suggests applications of our technique to some well known Frechet manifolds i.e. manifold of infinite jets and manifold of smooth mappings.
