Summary: This study is focused on a class of discrete mechanical systems subject to equality motion constraints involving time and acatastatic terms. In addition, their original configuration manifold possesses time-dependent geometric properties. The emphasis is placed on a proper application of Newton’s law of motion. A key step is to consider the corresponding event manifold, whose dimension is bigger by one than the configuration manifold, since a temporal coordinate is added to the original set of spatial coordinates. Then, its geometric properties are determined and Newton’s law is applied on it, when no motion constraints exist. Next, the way of introducing time dependence in the geometric properties of the configuration manifold through a coordinate transformation in the event manifold is investigated and clarified. Moreover, similar time effects introduced through the motion constraints are also examined. Based on these and application of foliation theory, a geometric definition of a scleronomic manifold is then provided, accompanied by a set of coordinate invariant conditions. The analysis is completed by deriving an appropriate set of equations of motion on the original configuration manifold, when additional constraints are imposed. These equations appear as a system of second-order ordinary differential equations. Finally, the analytical findings are enhanced and illustrated further by considering a selected set of mechanical examples.