Abstract: The geodesic characteristic of equations of motion for nonautonomous constrainedmechanical systems is studied in the modern setting of global differential geometry. Anecessary and sufficient condition for the dynamical flow of a monautonomous mechanical system with geodesic characteristic was obtained with respect to aconnection on jet bundle. The dynamical flow concerning the non-autonomous casewas always of geodesic characteristic with regard to torsion-free connections. Thus the motion of any nonautonomous mechanical system with constraints can be alwaysrepresented by the motion along the geodesic line of torsion-free connection on 1-jetbundle, which is different from the case in an autonomous mechaincal system.