  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 semisprays on a manifold $M$ and their induced
vector bundle structures on the second order time dependent tangent bundle
$\R\times T^2M$. Then we turn our attention to regular time Lagrangians and
their interaction with $\R\times T^2M$ in different situations such as
mechanical systems with potential fields, external forces and holonomic
constraints. Finally we propose an examples to support our theory.
