A coordinate-free version of the approach to mechanical systems with non-ideal restrictions developed by Udwadia (2002) and Udwadia and Kalaba (2002) in a series of articles is introduced. Some of its properties are then reinterpreted in a general geometric setting in terms of orthogonal projections. A geometric view of other aspects of constrained systems, inspired by their insight, is also presented.Facultad de Ciencias Exactas