The aim of this paper is to geometrize time - dependent Lagrangian mechanics in a way that the framework of second order tangent bundles plays an essential role. To this end, we first introduce the concepts of time dependent connections and time dependent semi-sprays on a manifold M and their induced vector bundle structures on the second order time dependent tangent bundle RxT 2M. Then we turn our attention to regular time Lagrangians and their interaction with RxT 2M in different situations such as mechanical systems with potential fields, external forces and holonomic constraints. Finally, we propose an example to support our theory
