In the present paper, we consider the synchronization of a pair of deterministic, second-order dynamical systems whose states evolve on a connected Riemannian manifold. The "slave" system has access to the state/momentum pair of the "master" system and is driven by a control vector field designed to make the former system sync in time to the latter. The paper also presents a theoretical analysis of a modified control scheme aimed at limiting the "control efforts" needed to achieve synchronization. An alternative version of the equations is also introduced that represents the case that the master system's state is observable under additive-exponential disturbances.