Abstract In this paper, we present the general structure for the explicit equations of motion for general mechanical systems subjected to holonomic and non-holonomic equality constraints. The constraints considered here need not satisfy D'Alembert's principle, and our derivation is not based on the principle of virtual work. Therefore, the equations obtained here have general applicability. They show that in the presence of such constraints, the constraint force acting on the system can always be viewed as made up of the sum of two components. The explicit form for each of the two components is provided. The first of these components is the constraint force that would have existed, were all the constraints ideal; the second is caused by the non-ideal nature of the constraints, and though it needs specification by the mechanician and depends on the particular situation at hand, this component nonetheless has a specific form. The paper also provides a generalized form of D'Alembert's principle which is then used to obtain the explicit equations of motion for constrained mechanical systems where the constraints may be non-ideal. We show an example where the new general, explicit equations of motion obtained in this paper are used to directly write the equations of motion for describing a non-holonomically constrained system with non-ideal constraints. Lastly, we provide a geometrical description of constrained motion and thereby exhibit the simplicity with which Nature seems to operate.