The Hamiltonian of a holonomically constrained dynamical many-particle system in Cartesian coordinates has been recently derived for applications in statistical mechanics [G. R. Kneller, J. Chem. Phys. 125, 114107 (2006)]. Using the same projector formalism, we show here the equivalence of the corresponding equations of motion with those obtained from a Newtonian and a Lagrangian description. In the case of Newtonian mechanics, the general case of nonholonomic constraints is considered, too.