The use of non-Hamiltonian dynamical systems to perform molecular dynamics simulation studies is becoming standard. However, the lack of a sound statistical mechanical foundation for non-Hamiltonian systems has caused numerous misconceptions about the phase space distribution functions generated by these systems to appear in the literature. Recently, a rigorous classical statistical mechanical theory of non-Hamiltonian systems has been derived, [M. E. Tuckerman, , Europhys. Lett. 45, 149 (1999)]. In this paper, the new theoretical formulation is employed to develop the non-Hamiltonian generalization of the usual Hamiltonian based statistical mechanical phase space principles. In particular, it is shown how the invariant phase space measure and the complete sets of conservation laws of the dynamical system can be combined with the generalized Liouville equation for non-Hamiltonian systems to produce a well defined expression for the phase space distribution function. The generalization provides a systematic, controlled procedure for designing non-Hamiltonian molecular dynamics algorithms which can be used to generate nonmicrocanonical ensembles, stationary nonequilibrium flows, and/or the dynamics of constrained systems. In light of this new general analysis, molecular dynamics algorithms for the canonical and isothermal-isobaric ensembles are examined, potential difficulties are illuminated, and the limitations of previous theoretical treatments are elucidated.